The dirt cheapest way to get from DC to LA is likely to be by bus. Greyhound (greyound.com) is the major long distance US bus service.
Amtrak (amtrak.com) is the US National Rail Service and is likely to be a bit more expensive, but more comfortable.
As far as flying there are numerous US airlines that can get you from coast to coast. Check out a website like yahoo travel or kayak.com to see ticket prices for multiple carriers. Also Southwest (southwestairlines.com) doesn't put there prices on these sites, so you might check them as well.
While a bus or train is likely to be cheaper than a plane, note that ground based transportation will take at least 3-4 days to cross the US--and trust me by the third or fourth day you will be ready to get off that bus/train. A plane flight will take 4-5 hours (realistically it will suck up a whole day) and on the whole it will be much more comfortable. Especially if you're taking a vacation I'd recommend paying a premium for a flight.