With four weeks, it actually shouldn’t be too bad to hit up both coasts. Especially since most of what you want to see on the west coast is actually within driving distance of one another (thorugh driving LA to SF would be a bit rough but I’ve done it). NYC can get really cold around now, but LA can be really hot, Las Vegas is a desert so generally hotter in the day and a more dramatic shift to cold at night, and a famous saying goes something like “the coldest winter I ever spent was a summer in San Francisco”. I went to college in LA and there were days in November/December when the weather was 80/90+ degrees, other days 50.
In LA, I’d probably suggest going to see the Getty Museum, Venice Beach, maybe find your favorite celebrities’ stars, see a play, maybe even see a TV show you like being filmed (depending on which ones you like. I’ve sat in a few studio audiences in my college days – they offer free ones but they do often require you to wait in a long line and you may not get in depending on which show so plan accordingly if that’s something you’re interested in). Korea Town’s got some of the best bakeries in the city and is close to where a lot of the concert venues are if there’s a band/performer playing in LA when you come that you want to see. That’s another thing – Korean food in this city is without peer, but coming from SD where we’re so close to Mexico border wait times are included in every traffic report I gotta say LA’s Mexican food is generally subpar. Though relative to Australia’s versions, it might be excellent.
I don’t really know if you’d be into this, either, but LA also has two basketball teams, a baseball team (you’d be a little too late for this if you came literally this time next year), and a hockey team that just won the Stanley Cup.
There’s tons of shopping in LA, if that’s something you’re interested in. Rodeo Drive is worth a peak just because of how outrageous and extravagent it is. I’ve never actually been, but I’d pass it on my bus route so I saw a lot of it and it’s amazing even from a distance. Obviously, there’s also less expensive shopping options available in other parts of LA. Whenever Australian friends came to visit my family, they’d stock up on various things since apparently a lot of stuff is a TON cheaper here. This may not hold true in SF or NYC, though, as they are notoriously expensive cities. This may have also been in the days before airlines charrged for check luggage, though, so I don’t know if it still holds true.
Not technically in LA, but within ~an hour’s~ drive is Disneyland/California’s Adventure. Definitely recommend them, they’re decorated for Christmas this time of year and gorgeous! However, if you don’t have a sense of nostalgia/magic associated with Disney, it might not be worth it to you with the crowds and the expensiveness. I’ve had passes to the parks for years but I know you couldn’t pay my parents enough to deal with going now that their kids are grown. You could also hit up Universal Studios, the new Harry Potter land/WB tour might be open by the time you get here, things like that.
On the more logistical side of things, the water’s likely to be far too cold to go in the ocean without a wetsuit this time of year, so be prepared if that was something you really wanted to do. For LA, seriously consider renting a car. It’ll make your life so much easier. Otherwise, you’ll probably want to do cabs. I got around using exclusively public transport, but it’s not ideal and will likely require something like Google Maps on your phone. Though I should say LA public tansit system’s not nearly as bad as anyone will try to tell you. LA has a subway system and a fairly extensive bus system, but you’re already going to be spending a fair amount of time in traffic. You just probably don’t want to add to that the extra time suck that inevitably comes with any public transit system.
(Haha, this is way longer than I thought. Tons of people complain about how terrible LA is in San Diego where I live, so I never realize how much I miss until I get to talk about it like this)