I looked at a map of Route 66, the Great Mother Road (that's a good thing!). I suggest whenever possible you travel on it.
In LA you must visit the Getty Museum:
http://www.getty.edu/museum/I'm much older than you but I enjoyed touring LA, visiting streets mentioned in 50s/60s Rock and Roll songs (Beach Boys/Jan & Dean). They are places that sounded kind of hip and fantastic when I heard them on the radio as a kid. It was fun driving on them for a few minutes.
Mann's Theatre (used to be known as Grauman's Chinese Theatre) is a must, along with the walk of fame. Very touristy. Very fun.
As you make your new home in LA, let me just say there are opportunities everywhere in the U.S. to be inappropriately felt by women. So you should feel right at home.
Barry