We usually just drive until we are at least west of San Antonio. I don't think there are any major attractions (by our definition anyway) until you get further west. You'll start hitting interesting parks in New Mexico and Colorado. In far south Texas of course there is Big Bend NP but that's far out of your path.
What roads do you plan to take? For Colorado we generally get on I-20 to Dallas/Ft Worth then switch to US 287. 287 is an excellent low traffic 4 lane highway. We take that to Amarillo. There is a great Texas State Park just south of Amarillo: Palo Duro SP. Which is a good stopping point. We have also stayed at Caprock SP and Copper Breaks SP which are both nice to visit but nothing extraordinary. Palo Duro has the Texas State Theater and they do an outdoor play at night which is worth seeing.
Anyway after Amarillo we work our way around to US 87 up into the NE corner of New Mexico. There is a national monument north of Amarillo: Albates Flint Quarry but unless they have upgraded the facilities they only do tours a couple of times a week.
In that corner of New Mexico is Capulin Mountain NM and it is definitely worth a drive to the top. It is a large volcano.
Right after that you hit I-25 and will be in Colorado.
