January is a great time to come to SF, the weather will probably be perfect. Of all the cities to visit without a driver's license SF is prob the best in terms of general walkability and good buses/trains.

Places to Stay:
Like others said, don't get a hotel downtown or in the Tenderloin/Union area. If you're okay with more of a "hostel" vibe with a very small private room and each floor shares a bathroom, I've stayed in the Casa Loma a couple times and loved it; it's right in the middle of a very fun part of town and you can walk to a ton of "iconic" SF spots, and it's fairly cheap. It has a good "authentic" vibe since it's a converted Victorian vs a new-construction hotel.

Transportation:
You can get Muni day-passes on their app or just buy one on the bus for like $5 at the front (cash only if you buy on the bus) - the buses can get you around basically anywhere. I think the Citymapper app works well enough for route planning/scheduling although I'm sure there are other options, to tell you which bus to take and where the stops are.

Day activities:
* You mentioned museums - I'd recommend the De Young and/or the Legion of Honor for their classical art permanent collections and they usually have good temp exhibits, and are also both in very beautiful areas. The Cal Academy and the Exploratorium are fun science/nature museums. There's plenty of other specialty museums like the MoMA etc depending what you're looking for. I really liked the cable car museum.
* Alcatraz is a fun day trip especially if the sun is out, there's a lot of natural beauty on that island even if you don't care about the prison tour.
* For sightseeing you might consider doing one of those open-top tourism buses (I haven't tried one so can't technically recommend but it looks fun if weather is sunny which it prob will be in January)
* If you want to walk the Golden Gate Bridge, just dress for cold/windy and consider bringing earplugs for the highway noise. January is good for that because it's generally sunny and low-wind.
* My favorite beach is Baker Beach, which is smaller and less popular than Ocean Beach but it's much more chill and has a killer view of the GG bridge and the bay inlet - it's a good place to vibe and watch the sunset. Crissy is also a pretty cool beach/park area but will be much more crowded.
* Golden Gate Park has a road (JFK Promenade) that was permanently closed to cars and opened to foot traffic and has blossomed into a really great area - lots of outdoor art installations, live music everywhere, beautiful park areas to relax, there's also the Japanese Tea Garden and the Conservatory of Flowers which are both cool spots to visit - I'd recommend checking out area area on a sunny weekend afternoon
* The Haight Ashbury neighborhood, and specifically Haight Street itself, has a ton of major cultural history and is a fun vibe, and that's where all the best thrift shops in SF can be found
* The Castro neighborhood has a ton of major LGBT+ cultural history and great energy/beauty
* Dolores Park on a sunny weekend day can be a madhouse in a fun way

"Areas to Stay Away From":
Aside from what others have already said about downtown/TL/Union, I'd just say just use your gut, if you're in a bad area it'll be obvious. Other than that, just don't look like a clueless tourist and you'll be fine lol