If you own your own property, watch for one to come into range of what ever you would like to kill it with (.460 Weatherby to broom handle). If you hunt on some one else's property, be sure to get permission first. Lots of farmers will welcome you if you are courteous, don't leave gates open, "ride" down the fences, and leave a bunch of trash on their property. You may want to offer to dispose of the carcass also (Usually back down the hole will suffice)
A popular method of groundhog hunting is to locate an area populated by the animals, secure permission to hunt them there, check and abide by local game laws, then watch for them to exit their holes into the open and shoot them with a gun, calibre, cartridge combination that appeals to you. Lots of folks use guns with small bullet diameter and high velocity. (.17, .224, .243 etc). Choosing a bullet which will fragment instead of ricochet is also desireable, especially in populated areas. There is also trapping and poisoning, but I assume you plan to shoot them. There are a lot of very dedicated varmint hunters who post here, check the boards, they will be invaluable in getting you started.
best to you and yours...
ears-