I'll reiterate what everyone else said: FE7 is a good place to start. I think that FE8 is a little too easy/forgiving; the only Fire Emblem games I've played to completion are 4, 6, 7, and 8, and 8 was my least favorite of the four.
Bu...but but 8 has awesome implied twincest! We can't ignore that!
Hey, if I wanted to have some Fire Emblem incest, I'd just have Aira marry her cousin in FE4. For whatever reason, Aira/Holyn is one of the most powerful combinations in the game and incest is really useful in general in FE4. I think I need to go cry in the shower....
I'm a little weird when it comes to the FE GBA games - I've played each of them multiple times and (no joke) some of my first gaming forum escapades were arguing about FE characters and strategies. I was Fire Emblem-obsessed from around 2004 to 2007. And of the four that I've actually finished, Sacred Stones is my least favorite.
I'll elaborate. FE8 lets you level-grind on random maps (making it WAY easier and less risky than every other FE game ever); FE8 has 10-20 fewer units than FE6 or FE7, which is lame, but allows branching promotions, which are kinda cool sometimes; FE8's units are by far the strongest in any FE game I've ever even tried, which means all of 1-10 other than 2 and 3 (easier and less risky than every other FE game ever). I like Fire Emblem games because of their huge casts of characters (good replayability), huge potential for different strategies and team setups, and their great balance of challenge. FE8 was too easy, its cast was too small, and its characters are all RNG-proof powerhouses. If you compare characters of the same classes in FE6-7-8, different archetypes clearly have higher bases and averages with each game, with a few weird exceptions like FE6's Karel or FE7's Oswin. FE7 is my favorite balance of character variety and power - FE6 has too many useless characters and FE8's cast is too small and strong.
...Damn. Now I want to replay FE7 or get off my ass and try FE9. I didn't own a Gamecube or Wii until around 2 years ago, and I never got around to play more than an hour of Path of Radiance (which seemed pretty good). Maybe I'll give it a shot once I'm done with some other Nintendo stuff.