Howdy infer and welcome to the forums.

A lot of people use .22 centerfires for varmints. They are flat shooting with low recoil, and ammo is relatively cheap and easy to find. THe .22-250 has a little more range than my .223, but ammo is a little more expensive. It is capable of taking varmints at ranges of over 300 yards on a good, still day. .22 caliber rifles are an especially good choice if you plan on keeping the pelts. Varmint tip bullets will make a small entrance wound, then fragment so that the bullet doesn't make a big exit wound.

I just got a .25-06, and I really like it. It is flat shooting and has a lot of range on it also. Recoil is still manageable. I use it for shots requiring a little more distance. Since it is capable of shooting a 120 grain bullet, it bucks wind a lot better than smaller calibers. It is also a deadly long-range deer and antelope caliber. At close ranges, it will do more damage to hides, but it has a lot more knockdown at long range. A 25-06 will handle a wide variety of bullet sizes, from 75 grains up to 120. For a cat, I'd definately choose a 25-06.