Alamo in San Antonio TX for a historical stop and you will be surprised when you get a look at the place.
Palo Duro State Park south of Amarillo TX. Stay a couple nights and watch the dinner theater. This is on the I20 route.
Not a lot along I10 in TX, FT Stockton has an old cavalry fort that worth a couple hours.
New Mexico is loaded with National Parks along whichever interstate you take. On the I20 route there are a number of Indian Pueblos you can visit, the Taos Pueblo is quite interesting
Along I10 is Carlsbad Caverns an absolute must stop, it is the most spectacular cave in the National system.
Also White Sands is worth a few hours, it's best seen in the early morning or around sunset.
Gila Cliff Dwelling is something of a diversion north of the Interstate north west of Deming. Near Silver City is the City of Rocks State Park worth an over night stop.
We don't know where you are going in AZ so it's hard to make suggestions. Like NM it is loaded with interesting stops.