The popular varmint cartridges are 223Rem., 22-250, and 243Win. Couple other notables are the 220Swift, and the new 204Ruger.
Forget the 222Rem., and the 6mmRem. - both are good cartridges, but for all practical purposes they are becoming obsolete.
The 223Rem. would be the cheapest route. Ammo is plentiful at discount prices, including military surplus. However, the best varmint cartridge, IMHO, is the 243Win.
The 243 will do a far better job of bucking the wind at extended ranges than will any
22 centerfire. It's very accurate, and can be had in a number of varmint rifles from
any of the major manufacturers. Plus, it can do double duty if you must, as it can handle
larger game such as antelope or whitetails.