The Getty Museum would be a very nice way to spend your time, and it's (depending on traffic) only about a 15 minute drive from LAX.

A Greyline tour is always an option, too.

We're in the middle of a Transit strike right now, but hopefully it will be over by the 4th.

If you want to you could post your plans in FlyerTalk Community and ask if any locals want to join you. There's lots of us here and we're a pretty friendly group (watch out for Jailer though).