I started off with a Ruger #1 in 6mm Remington for white tail deer and a Remington 722 hunting coyotes at dawn as they returned to their dens but soon discovered the 22 Mag was just as effective at the same range and much cheaper to shoot.
The 22 Mag was perfect for prairie dogs too. However I was so impressed with the .222 Remington's accuracy.
Jack rabbits are an over abundince in Kansas but anything from .22 LR to 22 hornet eliminates them easily. 

My go to for varmint hunting has been a rifle in .22-250 for more decades than I can recall now? Up close, 100 yds. or less for small varmints I use a Ballard .22LR with an old JW Fecker 12x scope. But once it's past 100 yds., or larger coyotes the .22-250 gets the nod.
I did buy another varmint rifle in the form of a CZ American in .17HMR about a dozen years ago, with plans for it to do duty after 100 yds., and save the .22-250 for 200 yds. and further. But the .17HMR is pretty much maxed out in calm weather at around 160 yds. so doesn't do a lot more after I stop using the .22LR. It hasn't seen much use because of this, and what use it sees starts sooner at around 75 yds just to use it more.