I go from Florida to the western states almost every year. When I'm prepared to get off the road and explore, Texas and Oklahoma have a lot to offer. However, the direct drive through doesn't have that much to offer. Please bear in mind, much of the scenery requires a little hiking (or a lot) and June-July can be very hot in the desert. During the summer, I try to stick more in the higher elevations. In NM, north of I-40 has some mountains to keep it cooler and some great natural scenery. Sedona can be hot, but is definitely worth it. Instead of hiking so much, take a Jeep tour. Stay close to Oak Creek for scenery and cooler temps. Colorado is a favorite place of mine in the summer, especially around Silverton and Ouray. The train ride from Durango to Silverton is a great trip, but there is also a good one from Chama, NM to Antonito, CO.
There is a risk of trying to cram too much into a single trip, which can be tiring and disappointing. My advice is to be very selective about what you do on the trip out, spend as much time as possible on the most important locations and don't spend all your time in the car.