I've used one that's an app on windows 8. it seemed to work fairly well, and as long as you can get the game file, it should run it. But that's only if you use windows 8... I'll find it and edit the link in.

Hm. Apparently it's only on the store, it's called VBA8 and you can use an external controller.

For JRPGs, the Final Fantasy and Kingdom Hearts series have always been popular.  Each Final Fantasy is completely separate so you don't need to worry about jumping in too late in the plot; I and V are my personal favorites, both of which are available on GBA.  Kingdom Hearts has four titles on Nintendo handhelds, but the story is complicated and it might be difficult to jump in.

Related to that, The World Ends With You (DS) is an absolutely fantastic game made by the same company.  It has a bit of a learning curve, but the gameplay is very unique and the themes and characters make it very engrossing.

For obvious reasons I'm compelled to mention the Mega Man Star Force series, all three of which are on the DS.  It's not one of the better known/received Mega Man series, but the first game has some of the best character development I've seen in my experiences with the franchise.

And, of course, there's always Pokemon.  Easy to jump into, easy to find too.

Tales of the Abyss is on the 3DS. Admittedly, it's a little hard to track down a copy of it, but it's a great JRPG. Tales of Symphonia and its sequel are also for the Ganecube and Wii respectively, if I recall correctly.

~B~

*obligatory second for The World Ends With You because my gosh that game*

I use/used visualboy for GBA emulating. Had it on here for four years now and it hasn't messed me up. Actually playing the first (US) Fire Emblem on it now. So that one works if you have Windows 7 or under, I'd bet. 

I honestly have no clue how to get new ROMs or anything atm though. It's been four years since I've done that, easily. 

And I don't know JRPGs at all really. Final Fantasy VI is super good as well!

Did you know there was an official Nintendo GBA Emulator? It was called the Game Boy Player, and it hooked to the bottom of the Gamecube.