Been there, done that.
I flew from Florida to LAX (6:00 a.m departure;arrived about 10:30 a.m.), took advanced reservation Super Shuttle to the pier (San Pedro), and checked in RAPIDO! I was onboard the STAR about 12:30 p.m., ate like a pig, and then took a well-earned nap. My cabinmate woke me up when he arrived about 2:30 p.m.
Net, net: I advise going straight to the ship. The pier is commercial and like any other you will dock at anyplace else. LAX has any transportation that you could ever want! (check the 'net). I promise you that you aren't the first persons to need to get from the airport to the ship!
enJOY the Mexican riveria! Blessings on your voyage! Go watch the cliff divers!