Ys Copypasta

My PC is shit I can't run them.
You're so, so, SO wrong. All Ys games work perfectly with single core processors with speeds equal or above 1.60 GHz, 504MB RAM or above and integrated graphics cards. Of course you'll sacrifice graphical quality but Ys games were never pieces of art in the graphics department (even though they look good).

Should I play them in order?
No. All Ys games are standalone adventures and perfectly enjoyable when playing them out of order. Although:

-Seven is the best entry for beginners. Ys has wallcrushing difficulty (healing items are scarce, you can't pause or heal during bosses and overall high difficulty) and relies a lot on quick reflexes. Seven allows you to carry healing items and use them during bosses, has a dodge button and a more gentle learning curve asides from featuring party members and more traditional RPG features. While it features some recurring characters and small references to older entries the game's perfectly enjoyable to be your first and it's the most isolated adventure so far.
-Origin should be played after Ys I&II. Origin rings on nostalgia for featuring a lot of elements from the first two games.
-Ark of Napishtim should be played last. VI explains the origin of all the things Adol experiences in older games, everything will make sense if you play this one last.

What's the chronological order?
-Origin
-I&II
-IV
-III
-V
-VI Ark of Napistim
-Seven

Best releases for each Ys game:
-Ys I & II

Ys I&II Complete (PC): Needs some patching and fixes to get working on modern PCs but it's the best version.
Ys Chronicles (PSP): Same as the PC version but has new artwork and soundtrack. You can switch between two artwork styles (Complete and Chronicles) and three soundtracks (PC-88, Complete and Chronicles).
Ys I&II (PCE): Both games in one. Impressive at its time for being one of the first RPGs to focus on storyline, colourful graphics, CD quality music and full voice acting in cutscenes (all of this in 1989). If you're one of those retro hipsters you'll love this version.

-Ys III
Oath in Felghana (PC): Canon replacement for the old Wanderers of Ys game that was released everywhere. Insane upgrade from what it originally was and contender for best Ys game ever and best Ys soundtrack.
Oath in Felghana (PSP): Lower framerate, uglier graphics and worse controls, but bearable enough and still playable. Makes up its faults with a new difficulty mode, Gallery mode, voice acting in certain cutscenes, a brand new intro and a Double Boost feature that restores HP. Good alternative if you want to play this on the go.

-Ys IV
Celceta Sea of Trees (PSVita): Canon replacement for the old versions (Just like Felghana did for Wanderers). Brand new combat system, arranged soundtrack and all that shiz.
Dawn of Ys (PCE): Not developed by Falcom. Not canon but has better soundtrack, gameplay and graphics.
Mask of the Sun (SNES): Not developed by Falcom. Canon entry (for now) but is inferior to the PCE version.

-Ys V
Lost City of Sand Kefin (SNES): Shit sucks, don't bother. Not translated either.
Lost City of Sand Kefin - Expert (Ys V Expert for short): Still sucks but with higher difficulty to adjust to Ys standards.
Lost Kefin Kingdom of Sand (PS2): Shit sucks, don't bother. Not developed by Falcom and not translated either... but more enjoyable than the SNES version so it deserves a mention.

-Ys VI
Ark of Napistim (PC): Fan translated and the first version released for it, thus the most pure of them.
Ark of Napishtim (PS2): Ported by Konami with additions that overstay their welcome. Sprites changed to ugly 3D models, voice acting with questionable quality, new cinematics with questionable quality and something called Alma's Trials which unlocks poinstless stuff. There are cheats to restore old videos and have japanese voice acting. Still a pretty good port.
Ark of Napishtim (PSP): Ported by some company that wasn't Falcom. Faithful to the PC version (no PS2 additions) but suffers from loading screens every time you change rooms. Bearable if you runn the game off a Memory Stick but it may still get on your nerves.

-Ys Origin
Origin (PC): Master race exclusive, unless Falcom ports it to Vita or something.

-Ys Seven
Seven (PSP): PSP exclusive so far.
Seven (PC): Chinese port with higher resolution but still featuring PSP's aspect ratio.

-Ys vs Sora no Kiseki
Alternative Saga (PSP): Falcom's love letter to fans. Power Stone-esque brawler using the Ys Seven system + jumping. Features most of the Ys Seven cast + Chester and most of the Sora no Kiseki characters + Lloyd from Zero no Kiseki. Loaded with Falcom references and lots of assist characters from Falcom games, giant OST collection to play during battles from original compositions, Super Arranges or even normal versions.  Surprisingly comes with a Story Mode that isn't an excuse plot. Consider this if you're a fan of any of the series, Falcom or just want a short burst of Ys action.

Other games:
Ys II Special (PC): Developed for DOS a Korean exclusive Ys game. Ys II Special mixes the games with the anime storyline and creates a different, bigger and possibly better game. The soundtrack is also expanded, with Adlib renditions of the original tunes PLUS new tunes by soundTeMP (Ragnarok Online, Granado Espada, MapleStory, etc) and art by Lee Myung-jin (did a manhwa named Ragnarok, which would become the universe of Ragnarok Online). The game still has the bump system but includes an attack button.
Download here: http://www.digitalsloth.net/ys/games/ys/DOS_-_Ys_II_Special_(1994)(Mantra).rar

Ys Online: Call of Solum (PC): Generic MMO based on the Ys universe, not a great game in any way but if you're curious enough to try it here are instructions to sign up and download: http://seldane.proboards.com/index.cgi?board=ysgeneral&action=display&thread=6313

Ys Strategy (DS): A non-canon cookie cutter RTS game based on the Ys universe. Shit sucks, don't bother.