Did you know it's a myth that you can't get around Los Angeles without a car? I lived there for years and return annually for visits, and have never owned or rented a car.
How do I do it you ask? I mainly walk, and when the distance is too great, I take LA Metro. I don't understand why more people aren't on foot, because the weather is so appealing. It's also the best way to explore all this fascinating, beautiful city has to offer. I can never take in enough palm trees.
I won't lie; it's not a picnic riding the bus due to the congestion and number of stops; however, it's cheap ($1.75) and will get you where you need to go. The subway system is great, especially traveling to the downtown area.
If you plan to take multiple rides, I suggest buying a day pass for $7. You can also purchase a tap card for $1, which is re-loadable and makes for a fun souvenir. http://www.metro.net/
Walking is my favorite way to spy the abundance of street art scattered around the city. You may even learn where to buy important items like earthquake supplies while you are wandering.
The quickest and most economical way of getting to and from the airport is via the Flyaway Bus. It only sets me back $8 to ride from LAX to Union Station. http://www.lawa.org/FlyAway
I challenge you to get out there and pound some pavement to discover the real LA. How else would I have spied these cool street art pieces if I had not been on the sidewalk looking up (even though they had their eye on me)?