For someone who got into this through the FF XIII series, I would definitely recommend playing FF X first -- which is good for a lot of the same reasons, only X is better.

After that, I would play through the classic "story trilogy" VII, VIII and IX.

Now you can either agree to downgrade your graphics even more for more good stories (IV and VI) or play a broken game.

I know this one is going to be controversial, but as the developers themselves say I don't think FF XII is a finished game. The problem, as some people make it out to be, is not really the political story. It's that you can go 5+ hours without a single cutscene. Meanwhile most of the game revolves around incredibly repetitive monster killing with almost no respite. Like with Xenoblade Chronicles, I think the only reason it has 90+ on Metacritic is because most video game reviewers are too lazy to play past the first 20 hours of the game.

XII has got some of the best art design and world-building in the series though, so if you are a patient soul I would still recommend giving it a look.

The rest of the mainlines are more retro JRPGs (I, II, III, V) that will probably be most interesting to you just for completion, if you're not into 16 bit grinding and looking for the right NPC to tell you where to go next. Oh, and there's the MMORPGs (XI and XIV). But again, this is something very different you have to enjoy in the first place.

Otherwise, from the spin-offs, the most worthwhile one is probably Final Fantasy Tactics. It's very different, but it's got a really cool story.

If you end up enjoying FF VII, I would also recommend playing Crisis Core for the PSP, which is a pretty good prequel. Phew.