My wife and I drove Route 66 for our honeymoon and had a lot of fun. We flew into Chicago and rented a car for the drive back to L.A.
It will be more expensive to do it this way, but don't make hotel reservations in advance, except for the beginning of the trip. There are a lot of cool things to see and do along the way, and you don't want to rush to make your hotel and miss stuff. While there are a lot of people doing their best to preserve the old road, stuff happens and the history is slowly disappearing. Also try to do your travel during daylight as much as possible, we almost missed a really cool abandoned 30's era gas station in Missouri because it was dark. It was one of my favorite stops on the trip, and none of my wife's photos came out.
As far as the car... There are LONG stretches of Rte 66 where you will see no one the whole time you are on the road. There are parts, (especially in Arizona and New Mexico) that are gravel, and parts that are very overgrown with brush. Your car will get scratched! Take food, water, spare gas, and some basic survival gear just in case, you'll be sitting for hours if you have a breakdown. Make sure your phone is fully charged before you hit the road each day for the same reason. Cell phone coverage was pretty spotty when we did it, things may have improved since then (has it really been 19 years?!?) but don't count on it.
Rte. 66 parallels the old Rock Island Railroad for a while, so if you're a railfan, there is some cool abandoned trackage and whatnot to see also. Most of all, have fun! And let me know when you get to Santa Monica, I'll buy you a cold beverage and we can swap stories!
