First of all, welcome! Second, you have activated my special interest, so I'll put the tl;dr at the top:
Generally, I recommend to watch in release order the first time through. Try to get through the core UC (First/Zeta/ZZ/Char's Counterattack) before tackling alternate universes. It sets up the backdrop from which everything else is built off of. Not required, just the recommendation of a fanboy.
In more detail:
How much did you enjoy 0079/First Gundam? If it was better than "eh," I highly suggest following up with Zeta Gundam. If you have your doubts, feel free to jump down to Alternate Universes.
Universal Century
The core of Universal Century is First Gundam, Zeta, Double Zeta, and Char's Counterattack (feature film). I recommend watching these before delving into any OVAs. Zeta is the sequel to 1st, ZZ is the direct sequel to Z, and CCA is the end of the arc. All of the UC stuff that came after is either far future, or retconned.
ZZ is contentious because it's a huge tonal shift from Z despite it picking up hours after Z ends. It also suffers from (reportedly) having some characters in the back half switched around due to CCA getting the green light for production. Some say skip it. I think it's worth sitting through the first 12ish episodes to get to the meat. YMMV.
Sticking with production order, next up are UC OVAs 0080 and 0083, along with F91 (feature film). 0083 in particular I very much suggest waiting until at least you've seen Z. It's its own contained story, but there's little bits at the end that lose a lot of impact if you haven't seen Z for context.
F91 was originally supposed to be a series that instead got adapted into a movie. It shows. It sits in an odd spot, jumping forward in time a bit with not much recognizable from earlier UC. It spawned the very popular (but thus far unadapted) Crossbone manga, and a prequel manga (F90). If you're only watching, this can be safely skipped if you want.
Next comes Gundam Victory, another large jump forward in time where not much is recognizable. V is... rough. In more ways than one. It's not bad, but it is brutal. Not required viewing IMO.
The first Alternate Universes are up next, but since we are still talking UC, it's not a bad time to watch 08th MS Team. It puts us back in the One Year War, primarily in Southeast Asia. It's a fan favorite for multiple reasons.
And I'll put it here just to say I mentioned it: there's a live action movie called G-Saviour. It's bad, very bad. I've seen cosplayers with better uniforms. Far future UC once again, might as well be its own thing. Completely optional.
Alternate Universe
As a way to try and jump start the flagging Gundam franchise, Sunrise came up with three AU scenarios and aired them back to back: G Gundam, Gundam Wing, and Gundam X. Wing also has an associated OVA/movie (depending on the cut) called Endless Waltz. Short form:
G is 1v1 fighting anime, but with robots.
Wing is angsty teen team Gundam.
X is kind of like a post-apocalyptic UC where instead of one colony dropping, they made it rain.
They all have more pluses than minuses IMO. Easy recommends that can be watched any time.
Capping off this era is Turn A Gundam. It's beautiful. Hard recommend, but it's better when you have a few other series under your belt.
My legs are asleep. Will continue later. π