Fire Emblem is a personal favorite franchise for me and I've invested many hours in both NA and JP games. I'll just give my personal thoughts and opinions to all the games released for NA.
Fire Emblem (GBA) - Absolutely solid FE game. It does serve partially as a tutorial for western audiences and it can get dull playing through that part if you're experienced but everything else afterwards offers a lot of unique paths, a good story, many characters to recruit and play, overall just one of the best FE games out there. You can get it on the Wii U eshop right now if you wanna play it.
Sacred Stones (GBA) - This is the only FE game that stands alone in its own universe. No other FE games to tie in with the story or universe. I do say it's a good game and very fun but for me personally, it's one of my lesser favorites out of the series. For one, it's the easiest FE game by far even on the hardest difficulty. Also, the story wasn't all that intriguing to me but overall it's a good FE game and has some memorable characters. Also on the Wii U eshop.
Path of Radiance (GC) - I love this one so much. It breaks a lot of the FE tropes with focusing on a new perspective in the form of Ike. Not a lord or of royalty but just a simple mercenary and his ultimate character development. This FE game was very colorful and unique with its lore and characters this time around. Gameplay, it lacks a little in some areas and it's probably the 2nd easiest FE game to SS. But the story and characters really do make up for that. Sadly it's tough to find and super expensive online. Good luck finding a copy.
Radiant Dawn (Wii) - Massive difficulty spike engage! Takes place a few years after PoR but it really expands on the story and perspectives in this game. You are actually jumping between different armies and different countries this time around which has never been done in FE before. I really do recommend playing PoR first since there will be a lot of spoilers. Unfortunately, it does come at a cost with some balancing issues with some of its characters. On the other hand, gameplay has improved a bit and the story throws so many twists at you that you feel like you're in Game of Thrones. Also, IKE IS BUFFED! Also hard to find unfortunately but I did hear they were restocking them again but who knows how long that could last. Best of luck finding it.
Shadow Dragon (DS) - Talk about a disappointment. It's suppose to be a remake of the original NES game with Marth and his story. Should be simple but they honestly made it a bit "too simple." It's pretty bland and kinda lacks a lot of content as a remake. I'd say try it out and play through it once. But otherwise I don't find it having much replay value or anything interesting to go back and play it again.
Awakening (3ds) - Probably the one you'll get the easiest access to. I do respect this game for essentially saving the series with its sales. It offers A LOT of great gameplay mechanics to enhance the experience. While it's a fantastic game, I don't think it's one of the better FE games when it comes to story and characters. The story feels rushed at different points and the characters feel like they only have a few traits and don't offer much of a full fleshed out personality. Otherwise, it definitely is probably your best option as a new player. If you do enjoy this, I HIGHLY recommend you pick up AT LEAST FE for GBA on the Wii U or something.