Now I've asked myself which region should I choose? I'm relatively new to the Forgotten Realms, so maybe you can recommend some regions?
FR is a traditionally a high-magic setting, which is an extra wrinkle to worry about when running 5e (because magic items are potentially quite the game-changers).
And more importantly: Which Setting Book should I buy? The 3e Forgotten Realms Campaign Guide? Or better the 4e Version (which is still available as a printed version, which I like)?
Maybe a single Book about, e.g. the Moonsee?
Whichever way you go, you're going to find a lot of game statistics that are of limited use. 4e's 'Points of Light' setting philosophy and PC-focus made settings into backdrops rather than the whole point of the campaign, so they tended to have a lot less detail (or detail focused on one small campaign area, like Neverwinter), and more player options, story hooks, and the like - fine if you wanted a campaign about PC Heroes of/in the Realms, not suitable if you want to know /about/ the Realms and it's important/powerful denizens like Elminster and the like. If you want a definitive vision of a setting, with detail and major NPCs, earlier eds will probably be closer to the mark. In general, the earlier you go, the more you'll get detail, clear/original vision, and a solid picture of the setting, with plenty of maps, organizations, NPCs, and the like. IIRC, there was an FR box set in the early 90s, if you could dig up a copy (heck, if you can get a db of old Dragon articles, Ed Greenwood's "_________ of the Realms" articles were pretty cool). Later supplements often tend, at least a little, to assume you're already familiar on some level. So far, 5e supplements have been adventures set in the realms, that only show you the realms in the context of the adventure - the middle part of HotDQ, for instance, was a protracted road trip tour of one region.
TL/DR: The older your FR sourcebook, the better.