1. New Mystery of the Emblem - Very player-phase focused, great maps with solid mechanics, as well as a good overall war story. This is probably the best FE, but I wouldn't recommend playing this first or even second if you're looking to get into the series. You'll only really appreciate this one once you've played many other games in the series.
2. Thracia 776 - Great variety of maps with great gameplay. The story is very personal and has some of the strongest moments in the series. Again, not recommended as one of your first FE's, but this one is unmissable for any serious fan.
3. Three Houses - Only finished one route so far, but this one knocks it out of the park in terms of characters, supports and world-building. Quite a few interesting maps in the prologues and later game. The battalion and gambit system is a positive addition to the gameplay and I'd like to see it return. Could take 2nd place once I've finished all routes.
4. Fates: Conquest - Brilliant maps, great mechanics. Falls short with its dreadful story, characters and game balance, but the sheer strength of the gameplay here puts it high on the list.
5. Binding Blade - A simple war story that can at times feel bland, but makes up for it with great maps and a solid cast of units to use.
6. Echoes: Shadows of Valentia - Weaker on the gameplay side, but makes up for it with the great polish on the character/supports/world building front. I like the changes to the forging system too.
7. Path of Radiance - Another game in the series with a good story, characters and world-building but average gameplay. Probably the best game to start the series with.
8. Sacred Stones - Similar to PoR, a good game that doesn't really shine in any area. Maps are too easy and aren't very interesting, but are serviceable. The characters aren't bad, but they're not too memorable either.
9. Shadow Dragon - Pretty strong in gameplay, but it's just too bland in every other aspect. Really not good for someone who likes FE for its characters and supports. The main merit in this one is to play it on its hardest difficulty and have fun abusing the tools the game gives you (Wing Spear and Warp).
10. Radiant Dawn - I find this somewhat of a slog to get through. Part 1 sucks as your units suck and the maps aren't that great. I wasn't really invested into the story at any point throughout the game. Part 4 gives me bad memories of boring rout maps.
11. Blazing Blade - Nonsensical story; boring characters; too enemy-phase focused because many enemy units have 1-2 range weapons and those have 1 range usually have a Mage or something thrown in to spice things up. There's a handful of good maps here and there but by and large the game is filled to the brim with uninteresting maps where you throw someone bulky with a Hand Axe into the middle of an enemy group and watch them clean house.
12. Genealogy of the Holy War - Completely knocks it out of the park with its story, world and characters, but the gameplay is trash as you spend ages just moving units across a sprawling map with nothing going on inbetween. FE4 is a game that is more fun to plan than it is to play.
13. Awakening - Popular with newcomers, but the more you play FE, especially the older ones, the more you realise that this one is good on polish and little else. Rout or Kill Boss is the objective every chapter and it gets boring. Maps are just big open fields with little going on. The story is split into three chunks, none of which are particularly engaging. Characters are one-note with little else going on. Lunatic+ is a bad way to handle difficulty. Finally, huge enemy stat inflation because of pair-up means that eventually, only Robin and Morgan are worth using.
14. Fates: Birthright - Awakening but worse in almost every way. Bad maps, characters, world-building, gameplay, supports. I don't have anything positive, truth be told.
15. Fates: Revelation - Birthright but with maps that also have bad gimmicks that just slow down the game. Along with Birthright, this isn't worth purchasing - just watch someone else's playthrough and read the story summary online.