I’m in San Antonio, and if you’re in Austin, there is Hill Country Kart Club in New Braunfels. It’s a little closer than the tracks that David mentioned, but those are great tracks as well. All the people at Hill Country Kart Club are great and willing to help. Our local kart shop is Texas Karts and is located in San Antonio.
For you, you can easily take the advice of those above. For your kids, your 5 year old would definitely be in a kid kart, and there isn’t too much to worry from there. For your 7 year old, your best bet is to get started with a cadet kart (assuming they’ll turn 8 next year). HCKC offers 2 main classes for cadets, both with a good turnout usually. There is Cadet 1, which is more for the beginner, you can run the LO206 or Micro Swift in that class. The other class is the Cadet 2, which will primarily run the Mini-Swift in that class. This is the class that my daughter races in.
Feel free to reach out to me if you have more questions about the track and club there.
HCKC has a Facebook page and a website that you can visit.