Universal Studios, Six Flags Magic Mountain, Disney at Anahiem, Knotts Berry Farm. I've visited three of the four, all great days if you have kids or you're "kids at heart".
Beaches / seaside - Laguna, Huntington Beach, Santa Monica, Malibu. (Yes, add in Venice Beach too if you like). All very different, all worth seeing in their own way. As I write, I wonder how Laguna is after the (?last year?) fires.
Hollywood - La Brea tar pits, stars in pavement, Rodeo drive. Overall I found Hollywood disappointing BUT I'm not much into that sort of place / lifestyle.
"Out of town", .... the Angeles Crest Highway, Ski-ing at Big Bear, Palm Springs (I'm told - not been there), Joshua Tree, La Jolla and San Diego, the Mojave Dessert, Pacific Coast Highway, Julian and Cuyamaca, and Death Valley and the Owens Valley up to Bishop and Mammoth Lakes. That list gets progressively further from LA but if you're there for long enough it's worth getting out to any of them.
My view is "a bit biased" ... I worked for a San Diego company for a while and regularly travelled their LAX and a drive down the "405". I have cousins in Huntington Beach. I learn to Ski at Mammoth. But I have never been there for more than three weeks at a time and that was a little long for one trip with all the stuff that had built up back at home when I got here. I would echo other comments that say "you need a car to practically do most of these things" ... and suggest that on your first trip you stop the first night near LAX rather than having a long drive after an 11 hour flight (I think I saw a "Venice Beach" post somewhere - that is close to LAX).
Other things that I do in LA (but not LA specific ...) eat well at so many places that do excellent fish and other food ... Mexican, Red Lobster, Sizzler, Taco Bell. Visit book and computer stores to find out what's the latest and greatest and will hit the UK in a year or two (my excuse). In fact ... you might be making me just a tad jealous of your trip; I could just do with a Breakfast Burrito at Mammoth's main lodge this morning followed by a run on the Cornice Bowl.
