San Antonio has a 'real' downtown, as does Austin. I would go downtown and park near the Alamo. The tour of the Alamo is a must as it is the heart of Texas freedom and is cool. Then, go down to the river walk - it meanders through downtown, just at a lower level. You can get on one of the river barges and have dinner. Eat in almost any of the restaurants on the river, Tex-Mex being one of the favs for SA. If you are feeling extra touristy, go up in the Tower of the Americas, part of Expo '68, with the obligatory revolving restaurant on top. If you are feeling historical, you can follow the Mission Trail, a series of missions founded way back when to help convert the local indigenous population and tend to their spiritual needs. If you like golf, SA has many, many municipal and semi-private golf courses. My goto for serious Jeep work is Bikini Offroad (
https://bikinioffroad.com/). They only work on Jeeps, mostly Wranglers. I would not take my Jeep to a dealership, especially Red McCombs or Anciria.
There is a Jeep club in SA, but they are on Facebook, which I don't do. You might look for them to get ideas about trails in the area. Corpus Christi is where I go with my Jeep, specifically to the Padre Island National Seashore. There is a north and a south Padre Island, with a split in the middle. I usually go to north Padre and drive down the beach for several miles. You can drive on the beach, but not on the dunes. There are some motels on north Padre if you want to spend the night. Or, you can drive down towards Brownsville from CC to get to south Padre. South Padre is more of a party zone with several motels, bars, etc.
The USS Lexington is permanently moored in CC, next to the Texas State Aquarium. The Lex is a WWII aircraft carrier, which I found interesting. If fishing is your thing, there are several fishing charter places in CC.
Before you drive into Mexico, you might want to do some research. Most US car insurance doesn't work in Mexico, so they will try to sell you some at the border. Check with your insurer and see about getting a rider for Mexico. Do not take any guns into Mexico, they will most likely be discovered and confiscated. Don't even think about buying illegal drugs and smuggling them back into the US. However, there are several pharmacies in Matamoros and Nueva Laredo where you can buy prescription meds (made in the same factory as those in the US) for a small fraction of the price charged in the US. You will need a doctor's prescription.
One final note - Texas is BIG. It takes nine hours of straight driving to get from SA to El Paso on I-10. SA to CC is about 2.5 to 3 hours. Austin has about 3 or 4 times as many people as the city was designed for. It has one main street, I-35 which is usually stop and go. Before they built the loop around the east side of Austin it could take 3 hours to get through. If you do go to Austin, sixth street is the main party area with thousands of college kids from the Univ of TX-Austin. The SA police are pretty nice and won't hassle you. The Austin cops are a different story - don't mess with them.
Good luck and have fun!