Route 66 is designated as a National Scenic Byway in most of the states it runs or ran through. IL, OK, AZ, NM. It is designated as a state scenic byway in MO, and as an Historic Road in CA. You can get info on attractions in each state from that state's scenic byway program.
As others have said, only some portions of the original road remain, but in most states, a route across the state has been designated. Some info: IL has a lot of remaining road. Interesting places along the route, but some long distances between Rt. 66 "kitsch" like the Gemini Giant in Wilmington. In CA, only the portion of Rt 66 in the eastern part of the state exists. As you near Los Angeles, the route is obliterated by interstates. But the Western terminus is still there-Santa Monica pier at the oceans edge.
One thing that is both interesting and depressing is the number of abandoned and derelict restaurants, gas stations, hotels and other attractions of the 1950s/1960s era.
The Disney movie Cars ( the 1st movie, not the sequel) was inspired by and all about Route 66. There used to be flourishing tourist towns all along the route, but then the interstate came and people bypassed all the neat little towns and their culture. The local organizations and the Scenic Byway Program are trying to restore some of that culture and promote travel once again along these historic routes.