Public transport from LAX can be unbearably slow but at least it's cheap. A free bus runs between the terminals and deposits you at the LAX Transit Center where you can swap to a city bus. Free shuttles also run to the Metro Green Line Aviation Station where you can catch a train headed in the direction of Redondo or Norwalk. Trains depart every 30 minutes.
If you're not the public transport type private shuttle buses will drop you at your hotel door: it's quicker than the bus and cheaper than a taxi. You can also try one of the many car rental agencies at the airport where rates are the most competitive in town. Taxis are convenient but expensive. Shuttles limos and taxis serve the area's other airports. In addition the Burbank Airport has Metro and Amtrak connections. You can find several major car rental agencies at all the region's airports.
Although LA is definitely built for cars it is possible to get around on public transport. The city has four public bus operators: MTA which goes most places; Big Blue Bus which serves the Santa Monica and West Los Angeles area; Culver City Bus which does Culver City and the Westside; and DASH, a minibus system that runs four routes through downtown. Metro Rail trains run between downtown and Long Beach Union Station and Western Ave and Norwalk and Redondo Beach.
The city of LA sprawls over such a huge area that unless time is no factor or money is extremely tight you're going to want to spend some time behind the wheel of your own car. Despite the sheer volume of traffic the city isn't hard to navigate if you stick as much as possible to the major arterials. If you want to get down to the nitty-gritty LA streets grab a copy of the Thomas Guides map books.