I would ride to San DIego, then start on your mapped route. If you ride east out of LA, you will be in yucky-big-city-busy-roads-bad-drivers-land for a few days, with no bike map to help you, in an unfamiliar country. And there won't be much in the way of camping options.
I have ridden from LA to San Diego, after flying in to LAX.
It's about a 2 day ride from LAX to San Diego, you can follow the Adventure Cycling Pacific Coast Route, or you can just get a city map in the LA airport, head over to the ocean and find the signs that say "Pacific Coast Bike Route" and follow them south. The only bad part is getting through the area near Long Beach - about an hour of scary-america. Just make sure it's daylight when you are there. The rest of it varies from ordinary city riding to quite pretty coast riding. There are plenty of services and a mid-point campground.
Also, when you arrive in Los Angeles, if you can afford a hotel room, get a hotel by the airport - many hotels have free airport shuttles, and many of those use a big van as the vehicle (of course, you should check with the specific hotel). That way you don't have to unpack and ride out of the airport after your very long international flight. Just shuttle to the hotel, sleep, put your bike together in the room, ride when ready (the next morning).
There is a store called REI (REI.com) that has several locations on the coast, including one in Manhattan Beach just south of the airport, that was helpful for buying stove fuel and the like.
If you want, you may send me an email anna_levine (at) yahoo (dot) com and I can mail you a copy of a few pages from a cycling guide book I have, that gives detailed route instructions with camping locations.
Have a wonderful trip!
