As a coach, I do not recommend having an air track. We have kids get injured on those at home and then develop mental blocks on those skills in the gym. It can be so frustrating for everyone involved. What I do recommend:
1) A pull up bar. Pull ups and leg lifts are great!
2) A panel mat for hard surfaces. If your house is all carpet, you can probably get away without one.
3) Floor bar. Great for squat ons, kip drills, press/stalder work, and eventually handstands and pirouettes.
4) Foam roller for aches and pains, and for recovery. Can also be used for overstretching.
5) Jump rope. Cardio and leg strength, and fun too!
6) Hula hoop - for older athletes, this can be weighted. When done with legs straight and together, and arms overhead, it's great for handstand shaping. Plus it's a really fun core workout.
Our gym had a guideline when I was competing that we could only do skills at home that we could do on beam. As a coach, I now really appreciate that perspective and share it with my athletes as well. I was one of those kids who couldn't get enough so having ideas of good things to do at home helped me a lot. I like to give as many safe options as possible for those kids who just really want more.