Suppose I might as well bring my input since I was mentioned...
For starting points, FE7, FE8, and FE9 are my premier picks. They're all great in their own right, and aren't too overly challenging, save for FE7's Hector Hard mode, and Maniac if you play the Jp version of FE9.
Awakening is also a great entry point in my book, however it's gameplay is really quite bad comparatively to other Fire Emblem games, although the amount of versatility you have within your army is still quite fun. I recommend it mostly because of it's fun cast of characters, and the fact that it has drawn in many people into the franchise already.
If it's not your cup of tea, at least try one of my previous suggestions, which are much more solid gameplay-wise, and their stories and characters aren't bad either.
And, if anyone's curious, my favorite FE game is Shadows of Valentia, followed by FE9 Path of Radiance, followed by FE3/12 (New) Mystery of the Emblem.