TWFirst -- Are you playing the guitar while singing the lyrics (my head hurts, by the way)
I travelled on portions of old US 66 in California and Arizona. In California, it starts at the Santa Monica pier, and you can travel the old route all the way to San Bernardino. It is marked by historical Route 66 signs. There's a lot of gas stations and hotels and fast food. From San Bernardino, it pretty much follows Interstate 15, and then it follows Interstate 40 all the way to Texas. Basically, there are few services on the road in the desert, but there are a lot of motels, food, and gas along the Interstates (but no youth hostels that I know of). The motels are pretty cheap though.
You'll have fun, and it's a great way to see the U.S. Don't go in the summer, though (it's very hot). Definitely take time to see the Grand Canyon, and some of the other national parks. And enjoy L.A.!