Might need to move this to General FF since it encompasses the whole series but for now whatever.

Its pretty awesome playing through the series sequentially. Before the numbers swelled up, I use to play the series in order before the new installment was released.

First, I would recommend FFIII and FFV. FFIII is for all intents and purpose, feels like what they were trying to accomplish with FFI but lacked the skill to do so at the time. It introduces the job class system and it's a just a fun romp. It takes the best elements from I and II and makes a truly well rounded title.

FFV brings back FFIII's job class system but introduces the concept of sub-abilities so you can mix and match the classes strengths to build more powerful classes. While V is certainly not as strong in terms of storytelling as IV and VI are, the game does offer some of the best gameplay of the old school FFs. It's also one of the funniest entry's in the series with the possible exception of FFIX.

As for which versions to play...

FFII - Stick with Origins, granted you can't play the original version til you beat it on Easy Mode but II's leveling system tends to rub people the wrong way so technically this is a good thing for some. Easy mode is still no cakewalk, it just makes leveling easier and gives you higher HP/MP boosts, and makes leveling magic easier.

FFIII - Purist will say you haven't really played III until you've played the original NES/Famicom version. I say screw the purist and play the DS version (soon to be released on iPhone as well ) The main cast are no longer faceless Onion Knights but now actually have some names and a bit of a backstory. The game has been rebalanced so you get more out of some of the classes and the game also added special skills for a few of them as well to give more variety in battle. The 3D sprites are adorable and the upgraded soundtrack is pretty snazzy as well (Though I still feel FFII has the better OST)

FFIV - This is a tricky one... Most people will agree that you should avoid the English version on the original SNES. It's a poorly localized and ultimately lobotomized version of the game. It then comes down to the GBA port, the PS1 port, and DS remake, (there's also the PSP remake that's coming out but I'll stick to what I know).

PS1: This is the closest to the original Super Famicom version, same graphics, a much better localization from the SNES version, and it retains all of the skills and some of the sub-plots from the original version. The main thing it keeps is the games original difficulty, and let me be the first to say that FFIV is the last of the "hard" FFs. I like the original difficulty myself but more importantly, I just like playing the game with it's original sprites and music. Though the main problems with the game is that while it's localization is much better than the first foray into the West, it's still the worst localization of the good versions of IV. It also has some loading problems but not nearly as ghastly as CT. It's probably not going to be easy to find either.

GBA: A much better localization from the PS1 version and it also changed to more Amano inspired character portraits. The game also added two new optional dungeons and the ability to switch out your party members at the end of the game. In the original version, the story dictates whose in your party, much like FFII does. The problem is that the game wasn't designed for this so Squenix had to re-balance the last section of the game and makes the final chunks of the game kind of easy. The optional dungeons are fun though.

DS: New graphics and a real customization system. The DS version mainly keeps the original story and cast and updates everything else. It has new character models who are more accurate to the original concept art, voice acting, and a new system which allows you to keep some of the abilities from allies who leave your party. The game has an even better localization than the GBA version, at least imho. It has rebalanced the game and enemy A.I. making the game much more challenging, and closer to the original version. It drops all the extras from the GBA version. It also has new mini-games and sub-quests that weren't in the main game and features a few additional story elements that were left on the cutting room floor of the original game. This is my second recommendation for which version to play.

FFV - Play the GBA version, yes it has some alteration to the music and graphics (though people don't bitch as much as the VI GBA port) but it also has an excellent localization and makes the cast feel much more lively and endearing. It has a snazzy optional dungeon added with a pretty cool final boss, it does add four new job classes though I personally don't like any of them. Still the game retains the sense of the original game and any changes are made for the better. Faris no longer sounds like a stereotypical pirate, Galuf and Bartz are much more funny and the game introduces Gilgamesh which alone should make people give this title the honor of being one of the best in the series.

FFVI - This one gets a bit ugly when asking which version. Some feel the original Woosely version on the SNES is perfectly fine, and it does have some awesome gnarm and charm going for it. The GBA version has a much better localization and updated script. The game feels more like the adult tale it really is but most hate the GBA version cause the graphics color palette has been altered but more importantly, the music was altered to fit the GBA and came off not so great...

Example:

Original

GBA

You can hear the difference. The problem comes with the fact that VI has debatably one of the best OST's in the series, so its a shame to have to listen to a less than stellar version. On the other hand, I still feel that the GBA script is the best version of the script and did a lot to bring out the characters personalities as well as the story itself. So at this point you might need to either make a dire choice of which is more important or just settle for the ROM versions. The fan scripts lack some of the quality of the GBA script but it at least isn't as simple as the Woosely script. If you do end up liking the game, I would suggest playing all the versions at least once.

FFVII - Only two options but I can only recommend the PS1 version. I've never played the PC version but I've heard it's slightly better cause it fixes some glitches and a few grammar points.

The rest of the series don't have much options in terms of version. Hopefully you have a PS3 or PSP so it will be easier to get a hold of VII-IX. X-XIII will also be pretty easy to find. Hope this was helpful.