On another forum, after a long diatribe about choosing correct wire sizes, I said "... OR, just use 2-2-2-4 aluminum for everything".
It covers every base and it's very favorably priced. There's nothing wrong with aluminum wire at such a large feeder size. (some novices have undue fear of aluminum, but they are not versed in the full details/lessons learned.) You didn't aim to be at such a large size, but #2 is a pricing sweet spot, due to popularity and the still-negligible mineral value in the wire. It's also cheaper than ANY copper option, even 30A #10.
#2 aluminum gets you 90A legally, so it serves almost all needs.
You want 90A to a garage.
The reason is resale value. Many home buyers foresee electric vehicles, and will pay extra for a house already wired for big power in the garage. How big is 'big'? The new, emerging "gold standard" is 80 amps for multiple EVs to share using "Share2" technology. (the enabling tech was baked into the original J1772 EV spec, which Tesla also follows. It's really quite ingenious.)
Provisioning that, leaves you 10 amps "left over" on each phase. One phase you can assign to lighting. The other phase can power the mandatory "20A" circuit with up to six 15A receptacles (provisioning 180 VA per receptacle as a rule of thumb borrowed from the commercial Codes).
"That was easy!"
See, this is why I opened with a discussion of wire economics. I wouldn't want you to think I was proposing #3 Cu, which would cost a fortune.
Cable types
Your familiar friend UF is basically NM cable improved for use outdoors. Both are limited to 60C thermal. That means #8 UF is limited to 40A, and #6 UF is limited to 55A and that's the largest UF made. It is not possible to get a 60A feeder with UF. NM goes bigger, but it's not allowed outdoors.
For an outdoor run, you will either do direct burial at 24" of cover, or conduit all the way with 18" of cover (PVC) or 6" of cover (pricey RMC). Note that RMC needs 12" under vehicle pathways.
A popular direct-burial cable is "MH Feeder" (Mobile Home Feeder) which is allowed underground but not indoors (unless it's in conduit indoors). 2-2-2-4 is the most popular size of that, since most mobile home installations use it.
For all-conduit runs, you use individual wires of either THWN or XHHW type. You can get any size of any metal - again #2AL is a popular size. The ground only needs to be #6AL.
Your distance is not far enough to worry about up-sizing the wires for voltage drop.
Note that the lugs on the subpanel are themselves aluminum, because (as the lessons taught) it's a 'universal donor', playing well with both copper and aluminum wire. Cheaper panels also use aluminum buses.
Panel size
Running out of panel spaces, and being stuck on a project, is the #1 topic on here relating to panels. Spaces are laughably cheap when buying a panel. As such, we really really recommend spending the price of a pizza on way more breaker spaces than you ever imagine you'll use. If you ever build a shop, you go through spaces like water - 240V tools use them 2 at a time, and 24 spaces is hardly excessive.
Since it's an outbuilding, your garage needs a "main disconnect" at the outbuilding. Due to a loophole, this "main disconnect" can be up to six hand movements, i.e. 6 circuit breaker throws. (2-3 120V breakers can be tied with handle-ties, making them 1 throw.)
If you exceed the 6 throw rule, you'll need to retrofit a "main" disconnect switch. The cheapest disconnect switches made are circuit breakers (just due to economies of scale). Many 14+ space panels allow retrofitting a main breaker, which is the cheapest path to doing that.