I'd pick one place to go just to make it easy and relaxed.

If you grab a cab to the Kodak, theatre and all the Hollywood stuff, it's about $35-40 each way. Lots of stuff around there, all the stars, all the costumed actors around Mann's Chinese Theatre, all the shops of Hollywood & Highland, etc. You could swing by the airport In & Out on the way back, but just arrange for a cab to get back to LAX since I don't think there are many places with cabs or hotels/car rental places with shuttles around there.

Beach option is really nice too.

Enjoy your trip!