In brief, prairie dogs are the "tree squirrels" of the grasslands. No trees so they live in holes in the gound like mice, rats, and their other cousins. West of the Missouri River as it courses north is a nice place to start. East 2/3's of Ks, too much farming. Western 1/3, good. 1/2 Nebraska, except the sandhills. 1/2 (west) of SD and ND. WYO except mountains. What you are looking for is "short grass prairie." These are the "blacktail" prairie dogs. There is a White Tail p'dog and it is protected in some places. BEWARE! (fed slam...) CO is good. MT is good. NM and AZ are a little rough. OK and TX are great if you have access. Alot of private land and pay to hunt I am told. Wyo did not require a license last I knew. CO wanted about $100.- for non resident small game with stamp. Do your homework.
b). In SD the native Americans "market" on the reservations. Rose bud, south center of the state, is legenday. But bring money. SD is a "TOURIST TRAP" IN all caps... Do a web search for Rose bud. Rules change. This is independent land or so they claim, but protected by Uncle Sam... Real "can o' worms" ... Standing rock is the other. North Center.
c). weapons. One fellow used to put a loop over the hole of fishing line, "lasso" them as they came out, and play them like a fish with fishing tackle. Another fella told me he used a muzzle loader. If you get into "unshot" territory, they will sit and bark at you at close range. Slingshot? .22 pistol? BUT once they get shot at a bit, they learn. You won't usually get close enough for a .22 rimfire. Those airguns would be a bit under power too. .223 will get you in the game. Scoped and 1 inch groups at a 100 yds. minimum. From there it is what you want to make of it. Shots will appear at 500 yds plus, if you can see that far. Little rediculous with a .223, but you can try. This is where the .22/250, .243, etc. take over. LUCK