That's a good road trip which I've done before. If you take HWY1 from SF to LA it's a looong drive but scenic. Keep your eyes peeled for Bixby bridge in Big Sur it's featured in a lot of TV commercials.
I lived in San Diego for 9 years and here is a small list of stuff to do in LA:
Fathers Office that has amazing burgers and a great selection of craft beers, my fav.
Casa Escobar for really good Mexican food, great ambience too. For dessert head on over to Diddy Riese for ice cream sandwiches.
Runyon Canyon for a hike up to Mulholland drive, a rewarding view of LA at the top. Or just drive Mulholland drive and gawk at the amazing views and residences.
Venice beach, 3rd street promenade, Sunset strip, Rodeo drive, Santa Monica all have shopping, restaurants etc.
In 'n Out for good fast food burgers. Google the secret menu!