I find for cities where I know I'm going to be parking, that its worthwhile to research the cities subway/mass transit system. You can usually get quite complete information online that will at least give you a good background on where each train/bus route goes and stops and how that fits in to where I want to go. If I have at least that, then I'm more comfortable when I'm traveling and am in a better position to make changes if I find somewhere else I want to go - with a little help from a route map and/or asking someone along the way.
As far as your specific cities go, NYC and DC are really the only two places where you won't want to have a car, and in DC most of the major sites are within walking distance of each other.
The other cities you'll probably want to have a car. I looked into using mass transit in Dallas and it was not a user friendly system, and its a pretty easy place to get around in by car. I have used mass transit to get around Vegas, but only as a money saver because I didn't want to rent a car. I've also driven there and found it to be a very easy place to drive.
One other note, Route 66 was decommissioned 25 years ago, so trying to drive it can be a bit of a challenge. There are sections of the road that simply don't exist anymore, so at times at least, you will have to use I-40 (the road that replaced it in that area) some of the time. I'll also mention that the 22 hours spit out by an online mapping computer is very optimistic, to the point it doesn't reflect real world travel times, which will push the time up to about 25 hours. Although, I thought you were going to Vegas, which is going to add another couple hours as a detour from the route.