Here's a brief rundown on each of the main titles of the series, and probably the most available version of the game in question (some may appear as DLC in places, but I'm not sure where; maybe Wii Virtual Console & PSN are the best bets, nothing really on XBL). I've also added some other notable games from Square to this list which somewhat fit in with the JRPG genre.
But yes, the list:
Final Fantasy I & II (GBA)
There's a GBA game with both on one cartridge (with some extra content), but the first few games in the series aren't overly special by today's standards. The grand stories the series were known for didn't really kick in until the 16-bit generation. Nevertheless, surprisingly fun to play.... but probably not the place to start with the series.
Final Fantasy III (probably DS)
Not a bad game, but the low-poly models make it feel very dated. Probably not the place to start.
Final Fantasy IV (GBA or DS remake)
Here is where things really started for the series, at least in regards to the storytelling; simple compared to modern standards, but exceptionally well-done. First game to use the ATB system, and it holds up quite well despite being old. GBA version is sprite-based, so is closer to the original; DS remake uses 3D models and adds in some voice-acting, but has a similar issue to the FF3 DS remake (sprites look better than the low-poly models, though this one holds up better). Excellent starting point.
Final Fantasy V (GBA)
Probably the best iteration of the job system, and one of the most fun to simply play. Storyline is solid and enjoyable, but weakest of the 16-bit titles in my opinion. Not a terrible place to start, but later games mostly moved away from the model it used.
Final Fantasy VI (GBA)
Best soundtrack in any video game, bar none. No, I'm not exaggerating. When an opera scene is considered the highlight of the game, before high-quality sound outputs were even possible, you have something special. One of the best storylines in the series as well, a perfect blend of drama and humour that isn't too simple nor too complex. Probably the starting point for the more sophisticated/complex storylines seen in later titles. A frequent contender for best game of all time, something which other games in the series may also claim. Definitely another great place to start.
*Chrono Trigger (DS remake)
This game came out not long after FF6, and is neck-and-neck with it in terms of quality on all fronts. The storyline and tone is more light-hearted than the FF series, but not completely lacking in drama; great time-travelling story. Soundtrack was also astounding Also a major innovator, by introducing team-based abilities and new game plus mods. Not a place to start for the FF series, but a game that should be placed just the same.
Final Fantasy VII (PS1)
The most popular game in the series, that much is certain. The best? Debatable, but still definitely belonging in the upper-eschelon for the series as a whole. Things probably started to get a bit too convoluted around here for the series' storylines. Arguably the game which was single-handedly responsible for making the series (and JRPG genre) more mainstream. Most people started with this game, so there's no reason to say this is a bad place to start.
Final Fantasy VIII (PS1)
Not a bad game, but not the best received. Worth looking into, but probably not representative of the series as a whole.
Final Fantasy IX (PS1)
Nostalgia overload! This game is a homage to the earlier games of the series, while taking advantage of the modern improvements. Storyline is great, the main issue is that it's probably one of the least remarkable; more light-hearted as well, with a hearty dose of humour. Superb starting point despite being enhanced so much by nostalgia.
*Final Fantasy Tactics (PS1, there might be a PSP version)
Quite a popular spin-off, mostly for the gameplay. Storyline is notoriously complex and the original is said to have some rather hideous translation issues.
*Chrono Cross (PS1)
There's only one big problem with this game... it's the sequel to Chrono Trigger, one of the best games ever made. There was no way in hell that this was going to live up to expectations, but it certainly tried. Might be worth looking into if you enjoy the genre, but isn't quite as required as it's predecessor.
*Xenogears (PS1)
Frequently referred to as the "Evangelion of video games"... well, that might explain just about everything now that I think about it. A cult-classic in many circles, with the most convoluted/complex story of the lot.
Final Fantasy X (PS2)
The first, and arguably the best, of the modern titles; I also hear there's an HD remake on the way. Fully modelled & rendered environments, voice-acting throughout, and an interesting twist of the ATB system seen in all the other games in the series since IV. Probably the best starting point due to it being from the modern era of games, while still having strong enough ties to the older titles.
Final Fantasy X-2 (PS2)
Every FF title is entirely independent in regards to the storyline & mythos... until this game, the series' first actual sequel. Some can argue this is where the series started to go downhill, but it's not a particularly bad game. It's definitely one of the odder entries, with the party being entirely female and the whole game being rather upbeat.
Final Fantasy XII (PS2)
And so the major changes begin, FF12 bears a strange resemblance to MMORPGs and Dragon Age with it's combat (which still being FF-themed); a lot of micro-managing of the AI is required, but you're allowed to set up the default decisions. Potentially the longest (single-player) game in the series as well, including having one boss who will take several hours to kill (though you allowed to save mid-fight); there's also more sidequest than actual main quest. Storyline is more politically oriented than other games in the series, and has issues of being spread way too thin. Reportedly great... but only if you get into the nitty-gritty. As such, not a good starting point for the series.
*Kingdom Hearts I & II (PS2)
Disney + Final Fantasy = the surprisingly good Kingdom Hearts series. Worth checking out in any case, but both of the games in the series (no idea about the spin-offs) have really slow starts.
Final Fantasy XIII (PS3/360)
The only game for the current generation of consoles, FF13 has issues. It gets a lot more negativity than it deserves in my opinion, but a huge reason for this is well-justified: it takes forever to get its act together. You're three to four hours into the game before you even get a taste of the more advanced gameplay, and it doesn't fully open up until you're around 20 hours in; it's safe to say it has the worst starting section of the whole series. The storyline isn't too bad, if a bit odd due to how much jargon gets thrown around at first; the same goes for the characters, though one in particular seems flat-out bonkers. Not a good place to start in any case.
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Hopefully that helps sort out where to start.
