I usually say public transit here will get you where you're going, but not necessarily easily or quickly. The other cities you mentioned have fairly well defined and obvious centers for activity, but LA really doesn't. Downtown could probably be considered the center, but there are a lot of other areas that have almost the same importance spread out, maybe more importance to tourists, and so it's harder for public transit to cover every place one would want to go. There are also a whole lot of transit agencies other than the MTA, the main agency for LA County, for example, the area where you'll be is probably a lot better covered by Foothill Transit than MTA and many cities have their own as well like Big Blue Bus in Santa Monica. Getting from point A to point B often involves more than one of them and multiple transfers in the same agency, especially if you're covering long distances.
The easiest thing for you might be to take a Foothill Transit bus to the Gold Line which goes downtown, and from there you can probably get a bus or rail line to where ever you're going, Red Line to Hollywood, express bus to Santa Monica, etc. Which is a nice segue to answer another question, yes there are express buses. Metro Rapid buses run similar routes to other buses, but they only stop at major intersections, definitely a time saver. There are also some buses that run on freeways, such as the 434 MTA bus from downtown to Santa Monica. MTA trains all stop at every stop, but I imagine they're usually the fastest if you're going to a place along the lines.
The day pass is definitely the way to go if you're on MTA. I don't think it helps for other agencies, but check into the transfer rules if you know which ones you're dealing with. There's a trip planner at mta.net that will figure everything out for you, though I've found I'm sometimes better off doing things myself with the maps and timetables. (I think it tries too hard to match the exact arrival or departure times I provide while I'm usually flexible on the time and more concerned with the other details) So in summary, public transit is a viable option, if you're ready to deal with some complexity and a longer trip than driving. Once you've got it figured out it can be easier and less stressful than driving and parking especially since you probably won't know exactly where you're going when driving either.
If you want to go to Orange County or somewhere else out of MTA land though, you're dealing with a whole new game and I don't know much about it. Metrolink is commuter rail for greater LA so that's an option for longer distances, though service is usually limited on weekends. For long distances on weekends I would have to recommend driving since there's usually not much traffic on the freeways on weekends anyway. I don't really know about cabs. I see a lot of them around the Sunset strip, so they're out there, but I haven't taken one to get an idea of the prices. My guess is that they may not be all that badly priced, but the long distances can rack up large bills anwyay if the cab will even go where you're going. Probably best for short trips like bar to nearby hotel.
Last I knew, passports were not required for crossing into Canada or Mexico. But I haven't driven over the border since about, the 1990s and there may be some new "security measures" I haven't heard about. Don't think so. I've never been to that part of Mexico so I don't know if it's worth going. Vegas is definitely an interesting place if you have time, and it's doable in a weekend just to see most of it (just make sure your AC works well if you drive
).