The main rifle/pistol I use for Rock Chuck shooting is the one I have in my hands.

I prefer the 22-250 for longer shots. I also shoot my AR as well, but you don't get the acrobatics that you get with a 22-250 or 308. But then again, with the 22-250 or 308, you can't keep 'em in your sight picture either, which makes it harder to watch the fun, or spot your miss.
They're pretty skittish and high tail it to their hole if you get very close, so you'll need a good set of bino's or a spotting scope. We usually run a two man team, one spotting, one shooting. Switch off after each kill or take the shot if the spotter can't get the shooter on the target. They blend in with the rocks and grass pretty well, so good communication and a good understanding of what you call "that white thing on the right" is. He could be looking at the whiteish brown thing on the left.