Note: I've only played FE7, 9, and Awakening.
I started with FE7 on GBA (simply "Fire Emblem" in the States), which is both the first US entry and widely regarded as the best place to start, and I'd say start there if you can. Each game brings something new to the table, but FE7 is not too simple or too complicated like some other games, and strikes the right balance in just about everything.
From there, well, it can get expensive. FE8 (Sacred Stones, also on GBA) is regarded as one of the lesser and easier entries, and most people will probably tell you to skip that.
FE9 (Path of Radiance, Gamecube) is one of the best games in the series, but it's incredibly expensive. It's also the first half of a duology with FE10 (Radiant Dawn, Wii), but I think FE10 is supposed to be a long, hardcore FE game that's more for fans of the series. I think it's also a bit pricey.
Then there's FE11 (Shadow Dragon, DS), which is a remake of the first FE with Marth - this one's supposed to be too simple and a step backwards in many respects. It's rather bland, but a used copy shouldn't be too expensive.
FE12 (Heroes of Light and Shadow, DS) was only released in Japan, and it recently got a fan translation. It's supposed to be pretty good.
FE13 (Awakening, 3DS) is the latest, although if you ask me, not necessarily greatest. It's got a ton of content and fanservice, and really throws in almost every feature that's been in FE games, but its map designs are a bit lacking (probably only noticeable if you've played the earlier games).
TL; DR
Hunt down FE7 if you can be bothered to do so, FE9 if you loved FE7 and you can afford it, then Awakening.