I recommend the following order, but haven't looked further back than the GBA era;
- FE(7, GBA): Blazing Sword. The introduction "Lyn Mode" can be kind of annoying the first time, but just keep at it. Also, this game can be brutally difficult at points and is kind of unforgiving if you miss stuff.
- FE6(GBA: translated); storywise, FE7 is a prequel to FE6. You can see where it all started, this game's mechanics are simpler at points (example; there are no other objectives than "seize throne), and it literally FLOODS you with units. It is fun, but not perfect, but I don't recommend it as a starter.
- FE8 (GBA); You could play this first or second if you want too. It isn't as long as the above, the story is isolated, and the class mechanism near perfect imo.
- Path of Radiance (NGC)
- Radiant Dawn (Wii)
- Shadow Dragon (NDS)
- New Mystery (NDS: Translated)
- Awakening
- Echoes
- Fates
eta: the 16 bit games never get old; it is what made FE big in the west. You can play all three seperately. The Radiant branch is really interlocked when it comes to story, the Cube game is the better one I tihnk, but the story can leave you dissatisfied.
Am on the fence when it comes to the DS games, though I think New Mystery far outshines Shadow Dragon. For the 3DS games, it is a toss up between Awakening and Echoes as to what is the better starter. Echoes is highly unconventional, but it does have a great focus on strategic aspects without being bothered by too many perks of the game. FE7 for example, if you are on your first playthrough, can have you branch off into bad decisions and making the game extremely hard in the end; promotion items and growth rates are hidden if you don't use guides.