Depending on where on the gulf coast you are, you may want to swing up into TN to begin with and take I-40 all the way out. There are tons of things along the way and the last half of it parallels (or becomes) Route 66.
Tennessee, you can start from the Smokies, through Music City and then stroll along Beale Street in Memphis.
From Memphis to Oklahoma City, you can detour down to Little Rock, but you really start to notice the terrain and vegetation changing. It start to get really dang flat after you pass through Arkansas.
In Oklahoma City, stop by the Oklahoma City National Memorial, especially at night, for the bombing of the Federal Building. Also stop at Cattlemans next to the stockyard for one of the best steaks you ever had.
As you roll into Amarillo, another big roadmark is the Big Texan restaurant and their 72oz free steak.
Albuquerque and the area have a number of attractions, You pass by some of the Pueblo areas.
You've got to stop in Winslow AZ and stand on the corner.
Trip up to Grand Canyon, Hoover Dam and Las Vegas for a little detour.
Barstow California is one of the last stops you'll find the first Del Taco and while you probably can't get on base, Ft Irwin is where a lot of Army desert training occurs.
Then over the mountains and down to LA.
Heading North out of LA, the Coastal route is downright gorgeous, but slow. A few places to stop would be Hearst Castle and then once nearer to San Fran, move back to it for Monterey Bay maybe Big Sur and Carmel. You can't miss stopping at the aquarium in Monterrey, generally considered one of the best in the country.
I've left out many places along the route, just mentioning a few.