Whew, there's so many options for things to do between Cleveland and and LA. A lot depends on how many side trips you want to take, and how far off to the side. but:

Lambert's in Sikeston, MO. Home of the "throwed rolls."

Arkansas and OK, at least on the route you're most likely to ride, aren't much. That's time to put on the miles.

The Texas/OK panhandle isn't much either. You can check out the largest (or second largest) cross in the western hemisphere in Groom, TX.
Rt. 66 through Amarillo is decent.

PowWow Restaurant and Lizard Lounge, on Rt 66 Tucumcari, NM

You could spend a week in AZ alone.
Painted Desert, AZ
Petrified Forest, AZ
Four Corners
Canyon de Chelly, AZ
Monument Valley, AZ

If you wanted to wander off the Interstate, during your trip, you'd find all kinds of things to do once you hit western New Mexico and beyond.