Strap in,
If you're more for the "role" side of "roleplay", you can't do much better than Burning Wheel. Character traits and beliefs intervowen into the very core of the system – you get better at fighting for what you believe, chasing your dreams. Fights are messy but rarely lethal, social interaction's precisely as important and impactful, magic is powerful and oozing with flavor, or, at the far lower end of things, could be your character's cobbling skills or big beard ends up saving the day. Anything they have is important. Simple d6 system where everything clicks together like good clockwork. Can't recommend enough. Setting is geared for old fantasy like Middle-Earth and Earthsea, but it's easy enough to port into other things: there's some scifi stuff to be found, or medieval Japan, or Dune with serial-numbers filed off, and the core books have plenty guidelines for whatever you yourself might want to turn it to.
If you're more for the "play" side of things, or just can't shake D&D out of your system, Basic Dungeons & Dragons has you covered, or many of the great OSR systems based on it (I like ACKS best). If you must play D&D, you should at least learn what D&D used to stand for. Go down into caves, avoid monsters, mind the traps, get treasure. Be cunning, be clever, cheat always, back down when you must. Simple as. It's like a medieval fantasy heist game, rather than the modern try-everything mess you may have gotten used to. Characters are defined by what they go through in-game, rather than tragic backstory and emotional baggage: see image.
(If that looks like a similar sort of exaggeration as in the whole "straights vs. queers" topic that brought us here in the first place… it kind of is, but the guy on the right is very accurate. Also "0e" stands for the Original Dungeons & Dragons, the White-Box, which is a bit too thick even for me. But maybe you'll get it one day. People still play it for a reason. Make it a quest.)
Or maybe you don't want to do a fantasy heist game? Maybe you don't know what you want to do and so need something that can cover a bit of everything? Pick up GURPS, go nuts. Run some dimensional crash where every party member comes from different worlds and different settings and have no idea what's going on and need to learn to work together. Most comprehensive chargen, fine-tune every single point, get some extra point from all that emotional baggage we all like so much. I like to just create characters for the fun of it, there's a lot to go for there. Not too intuitive to actually play, takes some time and dedication to chew into, but definitely worth your while: once you have it down, you'll never need to pick another system again.
Maybe something lighter? You haven't got the time to read all that shit and just want to pass a fun afternoon with friends? Or you're all completely new to this hobby and need to learn fast? Grab Risus. The whole thing's six pages long, but those six pages are worth the six hundred of some other systems. Grab an archetype, roll some dice whenever that archetype can fix things up for you, get bonuses for coming up with crazy outside-the-box solutions. Best introductory system in my opinion, hands-down.
Or you want to get spooked up for Halloween? AtlasJan brought up World of Darkness stuff, but I'd like to add to that Don't Rest Your Head. Can't sleep, go down the sewers into fairy realms, develop crazy-ass superpowers, survive the world of wacky shit without turning into a monster like them. Read Neil Gaiman's Neverwhere to get a good idea. Or pick up the My Little Pony hack, we all know how much we like the ponies.
You like pulp action? Flash Gordon, Indiana Jones, Doc Savage, raygun gothic science fiction and two-fisted adventure tales and wild western? Have a look at Savage Worlds. Fast-paced and fun, exploding dice everywhere, adventure cards make things crazy, great for improvisation from both player and GM side of things – though it breaks down a bit if you try for longer adventures.
For dark fantasy there's Warhammer Fantasy Roleplay, and its science fiction counterpart, Dark Heresy and derivatives. I'd go for the second edition of both: I've heard some contradictory things of the fourth edition of the former, and Wrath and Glory doesn't much interest me.
Got no friends but still want to roll dice? Check out Quill. Write letters for people, see how that goes.
There are other systems that I haven't personally tried but I've heard good things of. Shadowrun and Cyberpunk are different variants of the same setting, the former adding a bunch of fantasy to the mix. Call of Cthulhu I've only played little, but it seems to work all right for spooky adventures. Dogs in the Vineyard looks cool. Apocalypse World and its derivatives were never for me, but you might find things in them. Ryuutama is comfy as heck just to read through, but I don't know how well it works for practice. There's even some roleplaying games that need no dice at all, such as Nobilis, if you just really hate the idea of things randomly fucking up. For high fantasy crazy shit, Exalted should be named. And if you're just looking for something to laugh at with friends and the very concept of Content Warning gives you hives, give FATAL a read.
Bottom line – never play 5e. 5e is a terrible mashup of a system that tries to do everything and thus fails at everything. It's bearing the baggage of half a century of history, understanding almost none of it yet forced to hold on to it for the sake of nostalgia points. Throttled by the fat hands of the corporate overlords that believe in nothing and want nothing but your money. Forever steeped in controversy and drama that nobody sane should give a shit about. The sooner you drop it, the better off you are.
4e is fine if you're looking for tactical combat.
