The Fire Emblem series isn't exactly the most newbie-friendly like many other Nintendo franchises, but it's pretty fantastic on its own.
It's a strategy game with RPG elements thrown in at its core. You direct your troops to foes and whatnot, while being aware on how the map and enemies works, and choosing tactics accordingly. What made the franchise so famous is the permadeath mechanic. If a non-important character died, he's gone forever until you reload your previous save.
The best places to start are usually any Fire Emblem game that is released in English territories, but Radiant Dawn. That one is fiendishly difficult (if not unfair), especially from the beginning. Otherwise, there is no wrong place to start your first experience with Fire Emblem as a whole.
If you also own a Wii U, I recommend starting with the GBA ones first, as they're priced at $8 and are designed with newcomers in mind. Keep in mind that Sacred Stones let you grind pretty much anytime you need thanks to the world map, but the first English GBA outing actually don't.
There's also Awakening for your 3DS handheld, another great point of entry for newcomers. Granted, the story isn't good and that there's plenty of ways to end up with a gamebreaker if you don't mind grinding your characters (which can be done anytime like Sacred Stones does). The character development and the gameplay are what will make this game an enjoyable ride. If you dislike the idea of resetting your save file to undo your mistake, Casual Mode is a more convenient option that not only let you save anytime in a battle, but also dead characters will be revived on the next battle. By the way, there's a very short demo you can try it out on how to play the game and see if you'd like it, but it's not really representative to how the entire game will play.
Or you can just wait for Fire Emblem: Fates and pick the Birthright version first. Pretty much the same gameplay as Awakening does, but a number of changes are brought to the tables. Most notably, choosing your own fate at a certain point in the game, which can influence who would join your side, how the story would unfolds, yield you different items and some gameplay aspects. Going for Birthright will be pretty much similar to Awakening, while Conquest is more akin to the Fire Emblem games before Sacred Stones. The latter version is generally more difficult won't let you grind for exp from the map and resource is more limited. If you end up loving the game, feel free to purchase the other path you didn't went for and/or The Invisible Kingdom one. The game is only available in Japan at the moment, but it'll be out in overseas sometimes in 2016.
