Honestly, there's no "cheapest" airport, in my opinion. It's all up to the individual airlines, and what their costs are. LAX is the biggest, and I'd probably say it's an Aspie's nightmare for crowds and hustle & bustle and confusion.... Burbank (aka Bob Hope) is the most convenient (for me) and is said to be the quickest from when you enter to when you take off, and has the least security hassles and friendliest staff.
My wife uses Expedia to buy tickets, and you can choose what airport and compare prices. But there are a few catches, including that you pretty much have to be prepared to buy the ticket right away when you get a good quote from Expedia. I don't know too much about it, because she does it and I haven't paid attention really.
Charles