Depends on the kind of start you want. Want the "real" Fire Emblem of yore? Go with FE4(Genealogy of the Holy War) or FE5(Thracia 776), they are probably the best of the lot when it comes to the oldies but play rather differently. Thracia resembles more the "classic" Fire Emblem formula of challenging battles with creatively designed maps and battles, which include surprise reinforcements from the back galore and other such shenanigans. It's still one of the hardest games in the series despite the tendency of this franchise to pile on extra super hard modes of all sorts on their games, and features interesting mechanics like capturing enemies. FE4 is much more easy going and features the biggest maps of the series, if that is your kind of thing, it also pioneered the whole system of allowing the player to play with different generations of warriors that would be revisited later in the series.
If you don't necessarily want a classic experience but simply want to play what many consider the core essence of Fire Emblem, i.e. challenging tactical battles with no grinding and creative, though somewhat puzzle-like encounter design, then you do not really need to go that far since there are modern games that execute that formula flawlessly, such as FE12(New Mystery of the Fire Emblem) and Conquest, especially on the harder difficulties. You might wanna play these in the highest difficulty available from the get go, and then go from there as you unlock new difficulty modes, since these games are mostly about the tactical challenge anyway.
I also think Fire Emblem Awakening is not a bad starting point, despite it getting some flak, so long as you don't mind being given the ability to grind. I'll just repeat what I said in another thread:
Also, Fire Emblem Awakening is not too bad, but the game outside of Lunatic mode plays more like a min-maxing-oriented Munchkin RPG than a tactics game proper, and it is a lot of fun when taken as such, considering the brilliant multiclassing system and possible builds. That said, the game in Lunatic + mode is up there in tacticool goodness and challenge with the best of the series, such as Thracia, Conquest Lunatic, and FE12 Lunatic Reverse.
Word of warning though, the romances and anime tropes in FE:A are cringe-worthy, and it's not like that criticism can simply be brushed aside by saying it is optional, since one of the most fun min-maxing elements of the game(coming up with OP optimized children) is gated behind that aspect of the game.
But considering how much praise Conquest gets without being much better in that regard, I'd daresay people can look past that kind of thing and enjoy the game for the merits that I described so long as they understand it is not a conventional FE game outside of Lunatic mode. Whether it is a good introduction to newbies or not is debatable. If anything the game has the most complete all-in-one package of the series. You can start playing it in the easier modes where you can grind and min-max, get hooked and immersed, and then, once you understand the system, you can tackle Lunatic, which should put you up to speed with the hardcore tactical aspect of the series.
If you are able to stomach the game's presentation/writing, you are probably going to like it as much as any FE, and perhaps more since it is the most versatile experience in the series, successfully offering more than merely tactics, while also delivering that in spades (in Lunatic and +).
I think coming from dungeon crawlers you'd enjoy the class-change system in Awakening and the way it allows you to accumulate skills from your different class changes, though the game gives you a limited amount of slots for your skills like Stranger of Sword City. Also, the spell system in this series, and magic in general is nowhere near as interesting as in Wiz-likes, and there is also no chargen, not here and not (as far as I know) anywhere in the series, so in that sense it is more like Dungeon Travelers 2 where you have to plan builds with the characters you are given. But there are a lot of characters, so you do have a few options. Bear in mind that at the end of the day, even though character and squad building can be fun in this series, the focus has always been on tactical combat, so you wanna try getting into that aspect of it at some point in order to really know if you are into the series, since taken as a min-maxing RPG, though fun for a while, its appeal is bound to wear off, especially if you are coming from games that put much more emphasis on that and do it better. As I said, enjoy the game for its character system for a while, and then, once you have mastered it, try Lunatic(where there is no grind) and then Lunatic +, and get into the tactics side of things. Doing that you run the gamut of pretty much everything that the series generally offers within a single, coherent ruleset.
Echoes: Shadows of Valentia may be a bit of an eccentric starting point, but it does have some "dungeon crawling" elements, though exceedingly simplistic. The story is by far the most enjoyable of all the modern entries, if you care about that kind of thing. You'll perhaps be too spoiled by all the stuff the other games offer to enjoy this one's rather simple overall systems and design, so it might be worth trying first for that reason, as it is an enjoyable little game in its own right, and it has the best overall presentation.