The Final Fantasy games are many.  There are even more ports and revisions, some good and some awful.  Here is a list of (arguably) some of the best editions available for most games considered /vr/.
     
FF1: FF Origins on the PS1. Anything after just shits on it enough to make the extras worthless.  A patched NES version is also a grand edition worth playing and is better in some respects.  Stay away from GBA and later.
FFII: PSP version. Every revision fixes something.  If you're not in love with the looks try PS1.  The NES version is busted as fuck either way though.
FFIII: Pick. Both versions are extremely different but sure aren't any easier.  PSP doesn't have the DS' WiFi requirement for the strongest stuff. The original NES also has fan translations, try experiencing both if you want.
     
FFIV: Depends on how you want it.
>Purest
PS1. Closest you're getting to original without amateur translations.  It does however have loadtimes, which makes it slower than all the other options automatically as most of them have none.
     
>The most content
PSP has a nifty party arrangement and final dungeon for the original plus The After Years and Interlude (which is exclusive to PSP). Only thing missing are graphics options, but some like them.
     
>Hardest
DS for many reasons. Also has a New Game+ with a ton more extras that are only for it and a completely new translation.  This can feel extremely different compared to the others, because of the entirely different graphic style which isn't unlike III's DS/PSP version.
     
GBA is has the same postgame and ability to make your own final dungeon party as PSP with more classic looking graphics, but doesn't have TAY or Interlude.  Sound restoration exists, but there's nothing for the colors.  SNES exists but it either has an awful official translation, cut content, and easy as shit, or it has an even worse fan-translation than the official with all (make that most cut content; the translators actually censored some of the text) the cut content there.
     
FFV: GBA.  Best script, most jobs,and a neat postgame.  Sound Restoration exists, but no color fixes do.  The SNES version also has a translation but stay away from the PS1 version like the plague.
FFVI: GBA. http://vrshares.net78.net has probably the closest you're getting to a "best" as a lot of patches came out for it from color and palettes to restoring cut content.  It also has extra content and an (arguably) better and expanded script.  Many like the original but it's buggy as hell and has a love-or-hate localization.
FFVII: PC has a few translation fixes but has the same localization. If no mods are wanted (there are a lot, including graphic updates and a complete relocalization/translation) it doesn't really matter much.
FFVIII: PS1. PC doesn't have as much changes.  The PC version requires replacement files to play the original music, whereas Square fixed this for VII.  Note that in the PC edition you can play Chocobo World without the PocketStation.
FFIX: PS1. Has no other version.
FFT: PSP is good but patch that shit and find a bro.  If you don't have/don't want to do either, PS1 is perfectly great.  You can even put the newer translation with bugfixes on it.


Notable hacks and patches:
FF1 Restored (general all-over bugfix and enhancement, makes the NES the best version in some ways): http://www.romhacking.net/hacks/1631/

FFII NES Translation (You should probably keep in mind that NES II isn't the most fun experience, but it's still here and playable): http://www.romhacking.net/translations/139/
FFII Translation Addendum (Fixes graphic bugs): http://www.romhacking.net/translations/1565/ 

FFIII NES Translation: http://www.romhacking.net/translations/141/
FFIII Co-Op Patch (Want to play with a bro?  Try this!): http://www.romhacking.net/hacks/1797/

FFV SNES Translation: http://www.romhacking.net/translations/353/
FFV- Legend of the Crystals (Revision of the fan translation): http://www.romhacking.net/hacks/1788/
FFV Co-Op Patch: http://www.romhacking.net/hacks/2122/

FFVII PC Retranslation (this is recommended for a second playthrough as it's merged with a whole bunch of other patches you may or may not want): http://forums.qhimm.com/index.php?topic=14914.0

Samuel Slight's Final Fantasy VIII DLS (enhances the PC version's sound greatly): http://ffsf.aaron-kelley.net/download_8.html

FFIX ATB Gauge Speedup (Just what it says on the tin.  A lot complain about how slow the game is compared to the other PS1 editions so here you go): http://www.romhacking.net/hacks/1799/

FFT Complete (Fixes some bugs of the original PS1 Tactics and adds the newer PSP translation): http://ffhacktics.com/smf/index.php?topic=3659.0
FFT War of the Lions slowdown and stretch patch (Fixes up most annoying problems with the PSP edition.  Most slowdown is gone and the stretched screen is fixed.  They're individual patches.): http://ffhacktics.com/smf/index.php?topic=8490.0

FFIV Advance Patches (use what you want, just make sure you have an European ROM.  Original English releases were bugged to hell anyway.): http://www.romhacking.net/?page=hacks&genre=&platform=&game=817&category=&perpage=20&title=&author=&hacksearch=Go
FFV Advance Sound Restoration (The sound is now completely like the SNES version, not just music): http://www.romhacking.net/hacks/563/
FFVI Advance patches (In case you didn't like the vrshares edition.  Add what you wish.  Make sure to use an Euro ROM for the Sound restoration): http://www.romhacking.net/?page=hacks&genre=&platform=&game=819&category=&perpage=20&title=&author=&hacksearch=Go


Misc. notes:
-Note that the GBA editions don't add a ton and a half of content. Normally it's general weapon/item upgrades mostly for postgame and one huge dungeon along with much needed translation fixups (not with IV, it and PSP is mostly the PS1 translation). V is the exception because all but one new job is usable before the end, also IV has equipment for previous party members.  It is enough to make the game a definitive versions where nothing else came after.  In most cases the over versions are also viable if you don't enjoy the GBA's nerfs to looks and sound.

-Some spinoffs like Mystic Quest aren't included, all games not mentioned have only one version.  IX was noted because it's a main game.

-This list excludes the mobile editions. I, II, and Tactics are straight PSP ports, III and IV are pretty much straight ports of the DS version (III PC is yet another straight port of this), V and VI are for the most part (VI has one change to how Cyan's techniques work) the GBA editions with SNES sound and an honestly awful coat of new graphics (same style as Final Fantasy Dimensions if you're curious) and things you don't really need. /vr/ users usually are against these versions as they either add nothing, make the game worse, or you could run into problems depending on your device (slowdown, crashes, black bars on the side, etc).  If you must use a phone, most will tell you to just use a mobile emulator.

-Some argue that you should just play the originals on the console hooked up to a CRT as the creators intended.  If you wish to do this, you should probably ask /vr/ on how to start with grabbing a console/games/a flashcart in the case of II, III, and V.