I've played most of the FEs. For anyone wanting to get into the series, here's my input on it.
For me it's split into two main categories:
Classic Fire Emblem
NES: FE1, FE Gaiden
SNES: FE3, FE4, FE5
GBA: FE6, FE7, FE8
Modern Fire Emblem
FE Path of Radiance, FE Radiant Dawn, FE Shadow Dragon, FE Awakening, FE Fates, FE Shadows of Valentia, FE Engage, Three Houses, etc
The modern ones all kinda blend together for me. Some of them are better than others. FE Engage was the latest one I think and that one was pretty rad. Shadow Dragon is a super hard remake of FE1, while Shadows of Valentia is a remake of FE Gaiden.
If one wants to play the modern ones with anime characters and nice visuals, pretty much just jump in wherever. I'd probably start with the latest one and go back from there, since most of them aren't really connected.
If one wants to play the classic games that focus much more on deep stories and giant battles (that sometimes go on too long lol) with a more 90's fantasy aesthetic rather than anime, then I'd pick one of the three categories and do that set. The NES duo are super antiquated at this point but are where it all began and interesting to revisit. The SNES trio are debatably the high point of the whole series, with FE4 often spoken of as the best FE game. The GBA trio is my personal favorite part of the series, just be sure to play FE7 before FE6 as it's a prequel.
For what it's worth, if I ever do an FE series replay it'll be 1-8 and likely ignoring everything after that. I like how linear these games are and how it's impossible to get lost, which makes them ideal things to play during mindless downtime where one needs to relax.
TLDR for folks new to FE: For the full Fire Emblem classic experience, either start at FE1 or FE3 (depending on your patience level for NES gameplay) and go from there. The option of emulator speedup makes all of these a lot better. For the modern FE experience just grab any of the post-2005 or so FEs for your system of choice.