You can get from LAX to Union Station easily if you use the flyaway shuttle which costs $4. Yes, it isn't a typo only four dollars. From there you can take Amtrak to San Diego. Make sure to check the sign on the Flyaway bus, becuase if you take the wrong one you will end up in Van Nuys which won't help you. Look for the one that says Union Station. The bus stops are at every terminal, just look for the sign.
You can check out the schedule using this link. http://www.metro.net/default.asp
Good Luck. Enjoy your time in LA and good for you taking public transport. You are helping to reduce the impact on the environment.