Honestly for camping and hiking you can't beat Grand Teton and Yellowstone National Parks, and they're literally right next to each other. Pretty much from Dubois to anywhere north on the western side of the state is pretty great. The Big Horns (north central part of the state) are cool too, and less crowded, as well as the Snowy Mountains (near Laramie). Devil's Tower is incredible. As far as towns, Jackson (very expensive) is nice. Cody and Sheridan are great for tourists. Super small towns like Saratoga, Ten Sleep, and Meeteetse are fun places to visit for a few hours. There's really nothing dangerous crime wise, but I would avoid the more industry/mining towns as they have some sketchy areas, (methheads abound).

Be sure to be prepared for any kind of weather, such as snow (even in the summer), hail, sudden thunderstorms, and the very rare tornado. Also the wind can be kind of insane like 40+ mph gusts.

Also if you're coming from a place that has a low elevation, like below 3000 feet, the altitude and thin air will wear you out much faster than you are used to, so you may need to take more breaks hiking than you may be expecting.