Try to interairline you baggage when you first check in. Give your flight info to the person and they should be able to check it thru MIA as long as the first airline does interairline baggage.

Suggestions, the Getty Museam, via cab or bus; Manhatten Beach and walk up to Venice via cab; cab to Hollywood/W Hollywood if you want to do the very touristy walk of fame; or around Santa Monica as suggested above. I'd skip UCLA/Westwood since it's not all the exciting. You could also rent a car since they are cheap but might not be worth the time since the agencies are pretty slow (unless you have one of the club memberships)

Certainly don't stay at LAX!