One possibility would be getting a used APS-C DSLR from Canon or Nikon, both come with a wide range of their own and third party lenses. Since both are wide spread they’re comparatively cheap second-hand. That might vary depending on where you live though.
I’m not sure about the current state of things, but until a couple of years ago DSLRs had still an edge over mirrorless regarding wildlife and sports.
The current generation flagships and top prosumer mirrorless have action AF probably close enough to on par vs the previous flagship DSLRs. However, anything below that level may still be pretty slow for fast action.