Updated as of One Tamriel

Build tips:
What race should I pick?
Take a look at http://elderscrollsonline.wiki.fextralife.com/Racial+Skills to see what each race specializes in.
Next, choose whether you want to play a magicka or stamina build.
As long as you DON'T pick a race with only bonuses to the opposite attribute, you will be fine.
You can stop reading racial choice right now and skip straight to class selection, ESO is quite forgiving in terms of racial choice.
However, if you want to learn more, read on!

Because damage scales off your max magicka/stamina, you want to pick a race with bonuses to your choice of mag/stam.
e.g. Altmer stam? Worst. Argonian stam? Better. Redguard stam? Best.
Redguard mag? Worst. Argonian mag? Better. Altmer mag? Best.
Health passives provide some utility regardless of build choice.
For extra min-max, don't just avoid races without bonuses to your chosen attribute, instead pick a race with bonuses to your chosen attributes.
Class choice doesn't factor into racial choice, as viable stam and mag builds exist for all classes.

Current top tier race meta:
For PvE: Redguard (more sustain) or Khajiit (more dps) for stam builds, and Altmer (more sustain) or Dunmer (more dps) for mag builds
PvP racial choice is slightly different. Stam/mag regen/sustain is valued over max stam/mag (as you are constantly in combat), and crit is devalued (as impen aka crit resistance gear is common).

Racial Overview (from a mostly PvE perspective)
Altmer: t1 mag race (in both PvE and PvP), t1 mag healer
Argonians: t2 mag race, t2 mag healer, good tank, high sustain in PvP due to potion and healing passives. Swim speed is arguably the most useful "flavour" passive.
Bosmer: t2 stam race, highest stam regen passive for PvP, stealth bonus for PvP (one shot stealth archer is a thing in PvP, but not in PvE as it lacks dps)
Breton: t2 mag race, t1.5 mag healer
Dunmer: t1 mag race, t0 mag DK race (best mag dk race), t2.5 healer
Khajiit: t1 stam race (for PvE only; crit is somewhat devalued for PvP. Note that the Khajiit passive is weapon crit, not spell crit, and thus does nothing for mag builds)
Imperial: t2 stam race, good tank
Nord: t2 stam race, good tank
Orc: t2 stam race, okay tank, stam sOrc for max movespeed memes
Redguard: t1 stam race (in both PvE and PvP), good tank
Note: all t2 and above racial choices are more than viable for endgame content, no one will blink an eye if you show up for a vet trial.

t3 racial choice: picking a race that only provides opposite bonuses to your build choice, with no health bonuses (e.g. Altmer/Breton stam anything, Redguard/Bosmer/Khajiit mag anything).
t3 racial choices are still viable for endgame content, but it is harder as you will have to make up racial deficiencies via other means (champion points, gear, game knowledge).
I would not recommend picking any t3 racial choice if you are interested in endgame veteran trials (12 man).

Having said all this, every race/class combo CAN achieved 30k dps with the proper build.
So if you are still dead set on playing a Breton stamblade or Khajiit mag sorc, go for it.

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What class should I pick?
Every class has viable magicka and stamina builds.
Skills that cost magicka scale off spell damage and max magicka, and skills that cost stamina scale off weapon damage and max stamina
Ultimates scale off whichever stat you have higher.
While most class skills cost magicka initially, some can be morphed into stamina versions that cost stamina.
Take a look at http://elderscrollsonline.wiki.fextralife.com/Class+Skills to get a feel for what each class is capable of.
Damage types that scales off max stamina: physical, poison, disease
Damage types that scales off max magicka: magic, fire, frost, lightning

Nightblades: Sneaky cloaks, siphoning/draining sustain, assassin-y, magic and disease damage
Templar: 'Holy' Aedric magic, light/sun spears, healing, magic damage
Dragon knight: Tanky, 'earth' magic, fire breath and other dragon-y skills, the 'dots' class, fire and poison damage
Sorcerer: Conjured shields, blink (short ranged teleport), sustain (dark exchange/surge), pet summons (not particularly viable currently), lightning and physical damage.

All classes have viable dps, tank and healer builds.
DPS can be either stam or mag based
Tanking is typically stam based (as blocking consumes stam).
Healing is only mag based (no such thing as stam healer).

Current top tier class meta:
DPS: All classes can dps fairly equally, no top tier.
Tank: DKs make the best tanks (igneous shield, igneous weapons, unrelenting grip among others), then nightblades (mag "sap tank"), then templars, then sorcs
Healer: Templars make the best healers (restoring light skill tree), then nightblades/dragonknights, then sorcs

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I don't care about class or race, I just want to be the best meta choice for PvE:
DPS: Khajiit/Redguard stam any class, Altmer/Dunmer mag any class
Tank: Redguard/Orc/Argonian/Nord/Imperial Dragonknight (basically any race with some form of max hp/healing received/damage reduction/stam sustain bonuses)
Healer: Altmer Templar is slightly better than Breton Templar, and Argonian Templars are fine too.

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What should I put my attribute points into?
The current meta is all in stam or all in mag depending on your build choice.
When you are leveling if you feel you are too squishy you can put some in health, as respeccing is cheap.
You shouldn't put points in both stam and mag, pick one only and stick to it.

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For more in-depth build guides including bars and gear, I would look at Deltias (for newbies), Alcast, and the Theorycrafting section of Tamriel Foundry (look for guides updated to the most recent patch)

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General new player tips:
Start item trait research as soon as possible. Important traits are sharpened and precise for weapons and divines, impenetrable, and infused for armour
Researching time increases for each trait you know, get started early.
Deconstruct all weapons and armour that's not your equipment, not ornate and not for researching traits. This should be enough to max your blacksmith/cloth/woodworking skill lines before cp 160 (the max gear level) if not much sooner.
Sell ornate items to vendors.
Deconstruct all glyphs. Enchanting skill takes the longest to max.
Steal an item to unlock legerdemain.
Join the fighters, mages, and undaunted guilds (the guilds are located in the first city of each of the first non tutorial zones: Vulkhel Guard in Auridon, Davon's Watch in Stonefalls, and Daggerfall in Glenumbra).
Pick up lore books (glowing blue books) to advance the mages guild skill line. Kill undead/daedra for FG, and complete dungeons and get dungeon achievements for undaunted.
You may skip the tutorial zones via boats near the docks near where you first spawn. This is not recommended for your first playthrough.
Buy a horse (10k gold) as soon as possible. Start upgrading its speed first, then capacity, then stamina. It takes 20 hours to train one level.
How do I get 10k for a horse: if you steal from containers/lockboxes and only take green or above rarity items, you can sell them in an outlaw's refuge (found in most cities) and should be able to make 10k in 1 or 2 hours (dependent on your daily item fencing limit, which can be upgraded via the legerdemain skill line). Remember to crouch and be hidden.
If you get caught by a guard while you have a bounty and give up, you lose all stolen items. Same if you die to a guard while trying to escape.
When you get level 10, travel to Cyrodiil (default hotkey: L) and do the first quest. This will unlock rapid maneuvers, a skill that increases your move speed that can be cast while mounted, which you should use when traveling from place to place.
When you get level 15, you unlock your back bar. Put all the skills you want to level on this bar, and swap to it before turning in quests for easy leveling.
In general, don't spend gold on repairs while leveling since the drops you get will quickly replace your current equipment.
In fact, try not to spend any gold at all.
Things you should spend your gold on: Save 10k for a mount. Bank space upgrades (bank). Inventory space upgrades (pack merchant). Mount upgrades (stablemaster)
Try to equip at least one of each armour type (light, medium, heavy) to get experience in each armour skill line
All crafting skills are useful. Keep blacksmith, clothing, and woodworking research on one character (who will be your main crafter), as style motifs aren't shared between characters.
If you don't like crafting you can ignore it, just be prepared to have to find a crafter (guildmate?) if you need a crafted set made.
Hirelings passives are worth it. They have a chance to drop gold upgrade tempers, which are used for lategame, along with random mats of your crafting skill level.
Pick up alchemy reagents. They are the only skill line that doesn't have an associated hireling, which means the only way to get them is by picking them up (or buying crates in the Imperial City with Tel Var stones, a more endgame thing)
I wouldn't bother picking up provisioning ingredients early, as they are by far the biggest holders of bank space.
I wouldn't bother doing crafting writs until I had a substantial source of mats; I feel they are mostly an endgame way of getting gold upgrade tempers. The inspiration granted for completing writs does advance your crafting skill lines, however.

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FAQ

I screwed up my build. Is respeccing hard?
No, it's dirt cheap. There's a respec shrine in the major city of the second zone for each faction
You can respec all skills, just morphs, or all attributes.
It's dirt cheap, like 50g per skill point or 100g per attribute point.

How do I become a werewolf/vamp?
Either get bitten by a player who already has the disease at the werewolf/vamp shrine in the last zone of each faction (ask a guildmate in a large guild/zone chat)
or get attacked by particular werewolf/vamp npc enemies during an IRL full moon (low % infection chance)
or buy it from the crown store (don't do this, major waste of crowns)

What is CP?
Champion points, you start earning these via gaining XP once you reach level 50.
This is similar to the paragon system in Diablo 3.
Champion points are shared across all your characters; once they reach level 50, they instantly jump to your maximum CP earned
You can spend CP on bonuses (such as reduced stam/mag cost, more crit damage, more healing received, etc) in the CP menu.
The max assignable CP is 561.
The max gear level is CP 160.
The formula for XP required to gain one CP is based on max assignable CP (561); the lower your CP the faster you earn your next CP.

What is the difference between normal and veteran dungeons (4 man) / trials (12 man)?
Normal is easy mode, you should be able to complete them with little difficulty even with a non optimal build, and the corresponding loot dropped is less valuable.
Veteran dungeons/trials drop better loot, but are much harder, requiring a more optimized build to complete.

Why are my stats going down as I level up?
Since One Tamriel, everything is scaled up to CP 160. If you are using gear below your current level (e.g. level 10 gear while you are level 20), your stats will be scaled up less; thus lowered stats.

Will I run out of skill points?
No, you will have more than enough skill points to max out all the skills you need.

Where do I get more skill points?
Completing a harborage quest grants 1 skill point each.
Grab all the skyshards in each zone, including in each delve. Every 3 skyshards = 1 skill point.
Complete the main quest for each zone (each zone's main questline contains 3 quests that grant 1 skill point each)
Complete the group challenge for each zone's public dungeon (1 skill point each)
Each PvP rank grants an additional skill point.
Gaining levels up to 50 grants more skill points.
Completing the story quest for each dungeon (4 man) grants 1 skill point each
The final quest of the mages guild involves 2 skill points

How do I gain XP really fast and power level?
If this is your first playthrough, I strongly recommend playing through the story and paying attention to the dialogue and characters. The quest writing is above average for your typical MMO.
If the writing is not enough, you also get 3 skill points for completing the questline for each zone.
I would not recommend rushing to endgame if this is your first character, you will miss a lot.
Clearing a delve/dolmen/world boss for the first time also grants a burst of XP
Grinding tends to leave you with lots of XP, but a lack of skill points.
However if you still don't care about some soul-crushing grinding...
Well, grind spots tend to be kept secret.
Having said that, I think farming dolmens close to each other is not bad as each dolmen destroyed grants a burst of XP.
Alternatively, farm monsters in public dungeons, there are certain dungeons which are particularly packed with mobs.
If you have spare gold, craft all training gear to boost your XP rate, and drink Psijic Ambrosia to get an XP buff. Play during special events (e.g. Halloween, Christmas), there's usually a quest that gives you a doubled XP buff for the duration.
Being in a group with 1 other person grants a 10%(?) XP buff to both players, increasing overall XP gained - useful for public dungeon farming. However, having 3 or more players in a group shares XP, lowering overall XP gained, so don't farm in groups of 3 or more.
Download an addon such as MyXPView to see how efficiently you're gaining XP.

How do I make lots of gold? I'm not a new player.
https://www.youtube.com/watch?v=ghPBYukl92U

How do I get gold tempers?
Only from hirelings, crafting writs, or buying them from guild stores.

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PvP
The pre-level 50 campaigns are generally not very cutthroat (although there are some experienced players who remake characters for the sole purpose of playing in those campaigns)
The post-level 50 champion point campaigns can be daunting for a newbie, as some players have spent a very long time honing their skills and build, and have max CP. However, if you stick with a group and don't run off alone, you should be fine.
The post-level 50 non-champ point campaigns are not very populated.

If you're only interesting in unlocking ranks in the Alliance War lines by gaining Alliance Points, here are some tips:
Kill any delve boss in Cyrodiil for a 1 hour long 20% AP buff.
Follow a zerg and spam rapid regen/mutagen with a resto staff for the assist AP
Use your AP to purchase wall and door repair kits from vendors, and repair them for more AP

Get offense ticks!
Offense ticks occur when an enemy keep/resource is captured by your side.
The tick is equal to the the AP value of all player enemies that died within the vicinity of the keep/resource prior to the moment of capture.
The tick is then divided equally between all player allies within the vicinty of the keep/resource at the moment of capture.

Stick around for defense ticks!
Defense ticks occur when no player enemies have died for one minute in the vicinity of a keep/resource you control.
The tick is equal to the AP value of all player enemies that died within the vicinity of the keep/resource since the countdown started to when the countdown ended
The tick is then divided equally between all player allies within the vicinity of the keep/resource.

Note: you can get an offense tick, then a defense tick after if more enemies died after you captured a keep/resource.

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Useful Addons:
I use minion to manage and autoupdate my addons.
Skyshards: reveals location of skyshards on the map.
Lorebooks: reveals location of lorebooks on the map.
Minimap: shows minimap on your main screen. Some other UI addons have this functionality as well.
AI Research Grid: Tracks your trait research progress and motifs known
Lost treasure: Reveals location of crafting writ surveys and treasure maps you find on your map.
Mail Looter: autoloots your mail in one press and summarizes loot (hirelings, guild store sales, Cyrodiil PvP mails)
Destinations: Shows unknown locations of interest on map.
Dressing Room: Saves and swaps gear and skill bars. Useful for quickly swapping between dps, healer, tanking, questing, PvP, etc. setups
Srendarr: Buff tracker addon, shows all buffs you have active on main screen. Customizable.
Zolan's Junk Handler: Sets and remembers what items are junk. Autosells junk items. Autosell your ornates whenever you talk to a vendor!
Lazy Writs Crafter: autocompletes crafting writs if you have the requisite mats - no more manually crafting the required items!
FCO item saver: saves items under different labels, inventory filtering via those labels
Advanced Filters: further inventory filtering
Harvest Map: saves resource nodes you gathered on the minimap, can show pins ingame as well
Master Merchant: scans your guild stores to determines prices of all items. Requires you to be in high traffic guilds for accurate prices.
Awesome guild store: search improvements for guild stores
Kill counter: PvP addon, keeps track of your stats
Combat Metrics: useful for measuring DPS
Azurah: useful for unlocking UI frames. Also a UI addon.
FTC/AUI/LUI: various UI addons for showing more info
Wykkyd's: more UI stuff
Loot Log: Prints all dropped loot for your group to system chat. Useful if you're looking to trade for a particular piece of gear.
CDG Bank Stacker: auto deposits your inventory into your bank, as long as you have an existing stack. Won't make additional stacks.
Easy Travel: scans your group, friends and guilds list to fast travel to them. Much faster for than walking to wayshrines! More useful if you are in highly populated guilds.
Helmet Toggle: bind hide/show helmet to a hotkey
Innocent Blade of Woe: Turns off prevent attacking innocents if you use Blade of Woe, then turns it back on. Similar functionality for vampire feed.
Potion Maker: Allows choosing traits and making a potion from that
Multicraft: Craft multiples in one press, rather than having to press every craft
Crit Percent: What does 688 weapon crit mean? Shows actual percentage of crit
Mitigation Percent: What does 10k phys/spell resistance mean? Crit resistance? Shows actual percentage of damage reduction