If you like to party, you can't miss Billy Bob's Texas in the stockyards district of Fort Worth, just west of Dallas. Further west, Lubbock is the home of Buddy Holly. If you're into rock and roll, you are going to enjoy the Buddy Holly Center in the Depot District of Lubbock. Some of my favorite mexican restaraunts are little hole-in-the wall joints along the border in El Paso. If you like the out-of-doors, a slight detour south will get you into Big Bend National Park.
In New Mexico, a fair amount of the old-timey business of old Rt 66 still exists in Albuquerqe. Further west, if you cut south from Benson, Arizona, an hour will put you into Tombstone-the town "Too Tough to Die". It's a bit touristy now but it still a part of the old west history that is part of our heritage.A bit south of that, the old copper mine in Bisbee is a hole in the ground it will be hard to forget. Have a great trip.