You're in luck -- by late April the beaches should be open, and you'll be here on the weekend which is the best time to people watch. Here are some of my favorite things to do.
The canals of Venice and Venice Boardwalk (These two destinations are walking distance from each other but like two completely different worlds. The canal streets are so lovely and not to be missed).
The Getty Center (not to be confused with the Villa, though that's nice too). My favorite of the permanent galleries there is the one with the impressionist art but they'll be having some neat exhibitions while you're here. And even if you don't love art, the architecture of the Getty Center is beautiful and they have architectural tours, garden tours... I can't get my husband to stop photographing it when we go there.
A day trip drive up the coast to Santa Barbara -- the coastline is gorgeous and once you get to Santa Barbara, you can check out the Mission, the courthouse and if you have time and it's a nice day to be outside, maybe LotusLand. There are hiking trails near there, too. Just be sure to take the scenic coastal route on the way there -- on the way back when it's dark you can take the inland route which is faster.
The Hollywood Farmer's Market (not the Grove, the outdoor street market -- here's their webpage
Santa Monica -- people watching on 3rd Street Promenade and the Santa Monica Pier
I'm not a big hiker but I understand Topanga Canyon is really nice for that.
If you like classic movies, check out the American Cinematheque at the Egyptian Theater. There's an Academy of Motion Pictures museum in the works but it hasn't been built yet- they are only in the fundraising stage. You might also want to google film festivals, there are a lot of them held here.
For theater, check out reviews in local publications like the LA Weekly. Sometimes LA Mag will have something about shows too. The Broadway productions are typically at the Pantages and the Ahmanson (be wary of the least expensive seats at Ahmanson -- unless you're not prone to vertigo, you don't want to sit there). But there are a lot of smaller productions too. Goldstar
sometimes offers promotional prices on tickets.
If you like improv comedy, I recommend The Groundlings
Also there is a MeetUp group of Greater Los Angeles Vegan and Vegetarian Hikers -- they might give you some hiking recommendations if you contact them. Their page is here
There is a lot more to explore here including concerts, comedy clubs, Hollywood Bowl, etc. The Griffith Observatory is nice too. I'm running out of steam, but if you plan ahead you should have a good time.