I've shot at the Pawnee Sportsmen's Club and it's a pretty nice range but only goes to 200 yards. There is also a cowboy actions range, trap, skeet and five stand. It is on private land and not the grasslands though. This range is about 1/2 hour east of FtC on Highway 14. It has public hours and also a membership option that allows you to shoot whenever there isn't a organized shoot.
There is another members only range north of town that probably has a long waiting list and stretches out to 600 yards or so.
You can shoot in the grasslands too, but be sure to have a map and gps so you don't trespass on the checkboard of private land in the area.
Good luck with the move, it's a beautiful country with great weather and you can't beat resident permit prices for elk tags!