You can get FF1+2 on a GBA cartridge. It is a much friendlier version for newcomers than the separate originals. FF1 is available on the Wii’s VC I think if you want it the old fashioned way, but FF2 isn’t. FF1 doesn’t have any story, you just go into dungeons, kill monsters, take down the boss and move on. FF2 has a story at least. Its nevertheless still a tough game. They’re not that bad if you take them for what they are. Recommended with caution. I wouldn’t put this at the top of the list unless you want an old school experience.
FFIII DS is a controversial title. It has a very strange difficulty curve. This game does not have a lot of substance to it and I wouldn’t go for it unless you are aware of what old JRPGs are like and accept what you will have to do: grind grind away. Not recommended.
FFIV: get the DS version. I believe strongly that it is the best iteration of the game yet. The story is a classic, the music is good, the new graphics are impressive for the DS and there is even decent voice acting. I especially liked the changes they did to the gameplay, it made it feel fresh. SE released a sequel to it on Wiiware called FFIV: The After. I hear it isn’t bad. Recommended.
FFV: You can get this on the GBA. Its a decent title, although the game is not as good as IV or VI. I would play other games before this one. Low priority.
FFVI: You can get that on the GBA. Its a classic and the best of the 2D FF’s. Recommended.
FFVII: not much needs to be said, it brought final fantasy to 3D and to more people than the usual RPG crowd. Its a good game. Recommended.
FFVIII: Highly controversial. The game has a very annoying combat system. Most of the game is hit or miss, making it annoying to play. It has a few things it does very well, but parts that it does very badly. Not recommended. Hades, shut up, you know I’m right.
FFIX: good JRPG. They tried to go back to the fantasy setting of the 2D FFs and they made a lot of references. It was a pretty good game and a well rounded package. They did a good job with the music, the characterization and plot development. The leveling system , unlike FF8, proved popular and was retained in future games. Recommended.
FFX: My personal favorite in the series. Recommend.
FFXII: The problem with FFXII is that Yasuno Matsuno, the genius behind other Square titles like Vagrant Story and FFT, left midway through development and it shows. The game feels half finished. You can see where it wants to go with the story, but it fails. The pacing is bad, the voice acting is lackluster. The music is not Sakimoto’s best, but its good. The world is beautiful if you DO NOT have a high definition television and I loved the combat system / gambits (though some people felt it took the life out of the game by making things too automated, its a matter of taste). FFXII is single player MMO. Go into it expecting that. Recommended with caution.
FF Mystic Quest: SNES game. Was meant to be a game to introduce non JRPG players to RPGs by making it friendly and less grindy. Its not a bad game and the music is renowned. Its shallow, easy fun. Recommended.
FFTactics PSX: the gold standard of Tactical RPGs. Any new tactics game is compared to FFT. Great music, great plot. The PSP version has extras and a better script but the music quality suffers a little as a consequence. Recommended.
FF Tactics Advance: GBA FFT game. They took the Tactics Ogre engine and made an FFT game from it. One of the key games that got the whole Ivalice thing rolling. They made a nice world, but many found the gameplay slow and frustrating. It doesn’t have the depth of FFT. Its a shallow game directed at a much younger audience. Not recommended.
FFTA 2: DS FFT game. Very similar to FFTA. Directed at young players. Took a lot of hints from FFXII. Most of the game is about getting jobs at the tavern and doing them for your guild. Think of it a bit as an MMOified FFTA. The game is overall shallow, but it is pretty and the music isn’t bad. Not recommended.
handheld in order: FFIV DS, FFVI GBA, FFV GBA, FF1+2GBA/FF3 DS.
PS2: FFX or FFTactics (very different styles and gameplay), FFIX/FFXII/FFVII (light hearted fantasy vs not light hearted fantasy vs sci fi - whatever seems more interesting to you) , FFVIII.