Hi! It sounds like you're planning a fun trip. Early December is a very inexpensive time to travel in the U.S. so you should be able to get some good deals on hotels and rental cars (if you plan to drive). Later in the month prices do go up and rooms book up, so be sure to make your reservations early.
The U.S. has several good low-fare/no-frill airlines. You say you want to see "everything" - if the cities are close look into shuttles or buses, but if you plan to fly many airlines only serve a certain region.
Since you are starting out in LA I suggest Southwest Airlines (SWA) for cheap airfare - especially for regional flying, although SWA does fly nationally. A couple weeks ago I flew from Arizona to San Diego and it was $150-200 dollars cheaper on SWA than the other airlines. If you plan far enough in advance you should be able to get some good deals. Make sure you try to book a non-stop because some flights have three or four stops and you will feel like you are spending the entire day on the plane.
Frontier Airlines is considered low-fare and is based out of Denver, CO. It serves most major cities. AirTran is an eastern bases low-fare airline. Jet Blue is worth checking out, too. I would still price check American, United, and US Airways - depending on where you plan to travel.
Besides LA and NY what else do you want to see?