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Quest/event system:
So there aren't really "quests" in the game as per WoW standards.  There are story quests, Dynamic Events (DE), and Heart Quests (HQ).
Story quests are the main lore/story whatever, they always shoot up 2-3 levels in difficulty per, so you're usually kind of behind with them if you don’t do everything else.
Heart quests you find on the map (see map location system), they have a rep bar and you do random stuff to fill the bar.  Sometimes it’s just kill crap, sometimes it’s kill crap, rescue people, activate stuff, throw bombs at holes, punch spiders while disguised as a kitten.
Dynamic events are events in which you run around with a bunch of people and do dynamic quest stuff.  It is usually the fastest and easiest way to get exp, gold, and karma.  Monsters in these are scaled to how many people are in the zone and assisting.
World events are huge multi part dynamic events.
Aside from that, just scouting points of interest/waypoints/etc (see map location system) gives exp, and crafting gives a ton of exp (see crafting system).  ALL EXP IS PERCENT BASED, hence why you can still get awesome exp doing prior zones.
Hint: to give yourself some breathing room I advice doing THREE race’s start zones off the bat.  To access other start zones you go to your main city, through the Asura gate to Lion’s Arch, then through that hub to another zone.

Currency:
Karma is like rep points/tokens from doing DEs and HQs.  Karma is used to buy stuff from Karma vendors and HQ givers after you finish their thing.  In my opinion this is the best way to get gear outside of crafting.
DEs are dynamic events, big orange circles on the map lots of people do like crazy event shit.
You do stuff you get a tier of medal (none, bronze, silver, gold) and awarded with exp, karma and money
Your experience will almost exclusively come from doing DEs and heart quests

Map location system:
Points of Interest are just special subzones within a zone, like caves or wreckage or towns, these points do nothing but give exp/busywork and ways to coordinate with people via landmarks.
Vistas are like SWTOR jumping puzzles but a thousand times better.
Skill points are where you do some challenge (usually kill 3 guys, activate a beacon after killing 3 guys, do some zany thing like be awesome at firing mortars) and get a permanent skill point.
Waypoints are used to teleport between areas, for a fee that scales with your level (yay spending 3s per teleport in level 1 zones =/ )
When you finish a zone (100%) you get 4 random items (usually two weapons/armor pieces, 3 tokens for the mystic forge (see crafting system) and a bunch of a crafting mat, and of course gold and exp).

Weapon skill system:
Levels 1-5 are purely weapon skills
Every weapon type/combo/whatever your dude can use confers skills, each class has slightly different mechanics (like Engineers only get one weapon because their skills switch them to different kinds of abilities).
You kill w/ the weapon, you get exp for each skill unlock.

Skill point system:
Levels 5+ you get skill points, you can also get them from doing skill point map challenges (see map location system) and from 80 on you still get “exp” relative to 80-81 scale, but you don’t level so much as you get a free extra skill point.
Skill points are used to buy different utility skills/elite skills from your character pane.
Later on you can buy super rare crafting mats from the Mystic Forge vendor by spending Skill Points.

Trait point system:
Levels 11-80 you get trait points
Traits are like talent trees, except without trees because they are lines.  Hence trait lines.
Each line’s 5/15/25 pt tiles give static bonuses (like using X causes blind now too), and 10/20/30 pt selectable bonuses.
To get access to these, at 11 you buy a book from some guy in your start zone to unlock the 10s tier.  Then at 40 you get the 20s tier, 60 you get the 30s tier.
Hint: the long and short of it is you pick trait lines to set up a synergy with your chosen weapon(s) and gear up/gem out/sigil out (see enchantment system).
At 80 you can fill 2 trait lines and go 10 pts into another.
Sidenote: When you get scaled down to other zones, you retain your special trait skills, and sigil skills/buffs (see enchantment system) from items which makes you considerably more powerful than a standard guy at level 9 or whatever.

Scaling system:
When you go to earlier zones you get scaled down to I think about 2 levels higher at max than what you should be at the zone, meaning stuff is still a challenge.  Everything still gives a decent amount of exp (since it is % based, modified very lightly by your scaled level) and you get loot at random either for that level or for your main level, scaled down a bit of course to prevent 80s farming level 2 bandits (thus you still get lots of nice stuff from even beating up lowbies).

Skill area combo system:
Some skills have setup and finisher traits, listed in the tooltips.
Like if I use a fire circle area and then a whirlwind finisher I fling out bolts of FIRE, that cause burning status, instead of normal bolts.
You can trigger these off anyone's area, including your own.

Condition stacking:
This is more of a sidenote in terms of combat, but caused a lot of confusion early on.  When you fire off a bunch of the same condition (i.e. Applies Burning/Burns foe) it will stack the time it is on enemies.  However, many were confused as to how the damage "stacks."  It does not.  What happens is the time will stack but the largest damage-per-tick will be kept and ignore all lower ones.  Meaning you always want to open up with the biggest condition hit you have, then throw all the other ones on to keep the timer going.
Example:  Applying a 300dpt for 5s and then 2s later applying a 500dpt for 5s will tick 300x2 then 500x7.  However, applying the 500dpt first then the 300dpt next will tick 500x10.

Crafting/discovery system:
One of the biggest ways to get EXP is to craft.  However there is a catch.  You can craft normally, but most of the time just to level through it you have to Discover stuff.
You put base parts in plus an enhancer of some type into the discovery thing.  It guides your hand, too.  You can only select compatible stuff, and only things you haven’t discovered yet.  Also, something you have exhausted combos out of turn red and are unselectable.  Discoveries grant like 20x the exp and crafting progress as normal items.
Example: JC, you can make jewel settings and ring bases, then you discover a combo with a gem and boom.  You can also wrap a gem in a filigree to enhance it, then double dip and discover that enhanced gem with a ring base and setting.  For say, armorsmithing, you make an insignia out of cloth and monster parts, then make a lining and mold for a part and then discovery them together to get “Helm of the X”
At level 70, about 15 levels came straight out of crafting.
Also, you can always gather anything ever.  Speaking of which, nodes are unique to your character and nobody can steal your nodes.
The mystic forge is a place you dump a bunch of the same type of item and an enhancement or whatever and pray to the Random Number Goddess to get something really good.

Enchantment system:
There are a few main ways to add enhancements to items.  Usually an item will have a slot in which you can place a gem (jewelcrafting), a sigil, or whatever.

Salvage system:
Salvaging is what you do when you break an item down.  The % chance to receive an upgrade back does NOT apply to gems.  Otherwise you really want to use the 80%-100% ones to pull stuff out to use in other things.  Use the 20% cheap ones to just break down crap.
Certain items are labeled salvage items.  Those literally exist to just salvage into mats.

Bank system:
The bank is shared between all characters, and you have to pay real money (or pay stupid amounts of gold for real money currency) to get more tabs.  The big new thing is the collectable storage.  This is also shared amongst characters and is used to store up to 250 of any non-gold quality crafting mat in the game.
Sidenote: Gold quality crafting mats will NOT go to the collectables section of your bank, and are instead the ultra mega uber rare items needed to craft the best stuff in the game.  Store those in your bank.  I'm talking about things like Destroyer Slivers, Large Skulls, Globs of Ectoplasm, etc.

Loot tiers:
Grey < White < Blue < Green < Yellow < Gold < Red
Loot is unique to your character and everyone gets loot from monsters that drop it, so you can get free loot for just tagging something before it dies to someone else’s blade.  There is no “this monster is mine” crap from WoW.  Also, Green and up tend to have sigils on them automatically, with Gold (exotic) having the highest tier sigil.
As far as I can tell the only way to get Legendaries is to craft them using the 200 skill point item from the Mystic Forge, and a metric ton of other crap.

Set pieces:
Sigil "sets" are ways to customize your gear with set bonuses without having to deal with stats you don't like.  For instance, Sigil of the Centaur (X/4) means that for each Sigil of the Centaur you have on a piece of gear, you get the next step in the sigil's list.  You can salvage these sigils off of gear you don't like the stats of and slap them into ones you do.
In the final zones of the game (Melchor's Grasp, Cursed Shore) you end up fighting corrupted versions of the Human gods' avatars.  They are giant World Events, and when you are done you get access to their Karma vendors.  These vendors sell full sets of exotic gear for 42000 Karma for each piece.  This gear has the god's sigil set, so you can always salvage the sigil off if you don't like the stats of the gear piece.  Each god has different sigil set bonuses, and they are tied to the god as well.  For instance, Dwayna's is all about healing, and Grenth's is all about crit damage and spikey death dealing. 

World vs World system:
WvW(vW) is a weeklong contest of servers.  Three servers are pitted against each other for power points over four maps.  Three of the maps are power seats for each faction, with the other two attacking from the bottom, the fourth map is a neutral underground pit of traps and mazes.  You can move freely between each one but you do have to pay for repairs in this one and it is based off of your PvE skills and traits, though you ARE scaled to 80.
There are also lots of dynamic quests, like escorting and killing super NPCs, as well as the ability to build siege weapons like trebuchets, ballista, boiling oil, and rams.

Regular PvP system:
This is unmarried from your PvE character.  You have the same class, but you are set to 80 and have full skill access.  Any skills you buy specially in PvE do not carry over.  Also, race skills are not in PvP for balance reasons.  Finally, you get gear and skills by leveling up via PvP, and the PvP is in a small scale 10v10 rotating map tournament format.

Example synergy setup: my 2HGreatsword Guardian
Full power + condition (bleed, poison, burning) damage trait full crit chance crit damage trait tiny bit in HP trait
I have a thing called Virtue of Justice (VoJ).  When I hit it, it applies a fire damage DoT to my next enemy.  Traits apply blind to every enemy near me when I pop it, and it refreshes the CD when I kill something, and applies vulnerability (1% more damage to them) in an AoE.
2HGS abilities of note are AoE spin to win, an AoE chain yank to me, and a sword charge/flip smash.
My class abilities for right now are teleport-attack that applies fire, fire AoE ring
So, trait lines give me 10% more damage on burning enemies, 5% more greatsword damage, greatsword strikes heal me, 5% more damage to blind enemies, My usual combat flow is this: run in, aggro all bears
Fire dash to an enemy, spin to win chains, VoJ, fire circle, flip smash gives fire aura, spin to win bolts flings fire bolts.  Everything dies.