Any tips on places to visit in this beautiful state would also be greatly appreciated
Catalina Island, the 'Airport in the Sky', 20-25 miles off the LA coast.
Challenging the first time - and care needed every time - but a great experience.
As can be seen in the photograph, the ground drops away sharply from the threshold of 22, so it's important to keep the picture focussed on the numbers, ignore peripheral vision, and be prepared for a downdraft on short final. You don't want to undershoot.
The flare also needs extra care because you're landing on an uphill slope which can be deceptive and, just as you touch down, the runway disappears. (It's 3000+ feet but there's a hump.) Having landed, you can treat yourself to a buffalo burger in the airport cafe.
Getting to Catalina :: aircraft
LAX
The VFR corridor (LAX Special Flight Rules Area) is well worth doing.
4500 feet NW-bound and 3500 feet SE-bound, right over the top of LAX.
(Airline go-arounds are not above 2500 feet.)
Or, if you want to experience landing at a major international airport -
The landing fee used to be about $25 - less than at some small airfields in the UK.
San Diego
Flying down the coast from LA to San Diego is a fabulous trip. (About 100 miles). The two GA airfields are Montgomery Field about 6 miles from downtown and Gillespie Field at El Cajon about 10 miles NE.
If the controllers at North Island Naval Air Station and San Diego International (Lindbergh Field) are in a friendly mood and traffic allows, it's usually possible to get clearances to fly low level (not above 500') across the bay near the Coronado Bridge and then continue up the coastline. The San Diego bay area is spectacular - one of the most beautiful waterfronts in the world.
As you'll be in Southern California for a week, I recommend making it a day trip which would give you time to explore.
FL