HI! :D Thank you for taking an interest in this awesome series! I’ll point you to this question I got before, but since I never got this specific one I’ll reply anyway :D
You can play almost all of them in any order, although some are sequels (FE5 after FE4, FE10 after FE9) and, in order to better understand the story and returning characters, you should play those in release order :) There are other sequels and a prequel (FE7 was released after FE6, but chronologically takes place 20 years before), but you can enjoy story, characters and lore regardless of play order.
If you’re starting out and you have a 3DS, get Awakening: it has Casual Mode (units return even if they die) and helps you get into the mechanics. If you have a Wii U, you’ll find Fire Emblem 7 (classic game and has a tutorial mode as part of the storyline) and The Sacred Stones (has a world map similar to Awakening and skirmishes where you can grind) on the Virtual Console. GBA games are also very easy to emulate, and if you don’t mind dabbling in emulation, go here! :D
Afterwards, you could try FE6/Binding Blade – follow-up to FE7, has the same world and game engine as FE7/8 – or maybe try FE11/Shadow Dragon and its sequel FE12 (remakes of the very first FE game and its sequel FE3, starring THE ORIGINAL PROTAGONIST, MARTH!) – note that, being remakes of NES/SNES games, these are more ‘basic‘ in terms of story and lack some mechanics popularized by later games (Supports, Skills, etc.) – or you could move on to Path of Radiance (FE9) and its sequel Radiant Dawn (FE10)! FE9 and FE10 are among the most complicated FE games, with the plot dealing with wars, evil goddesses, shapeshifters and much more! (FE10 especially deals with rebuilding the nation YOU trashed in the previous game from the defeated’s POV, fighting against your characters on different sides of the battlefield, dealing with the issues of queenhood and more!)
Sadly though, both FE9 and FE10 are pretty pricey or hard to find these days :/ You can emulate them too, although those require way better PC specs than a GBA emulator – more on that here :)
…AND IF YOU FEEL UP FOR A CHALLENGE, THERE ARE FE4 (Seisen no Keifu or ‘Genealogy of the Holy War’) AND FE5! >:D These two are widely considered the HARDEST FE games and feature new mechanics – like the marriage/children system, skills, HUGE maps with multiple castles to conquer, capture system, etc!
The plot spans two generations and spins a dark, unsettling tale of war and tragedy, with blood, dragon gods and noble houses as protagonists, where your characters will become unwittingly and tragically entangled in a sinister plot to resurrect a dark god and plunge the world into chaos…
DOES IT LOOK IT COULD TURN UGLY??????????
YOU HAVE NO IDEA.