Lots of public land to shoot on, just follow the rules for shooting on public land;
Must be 150-yards from any roads, trails, campsites, or homes. And you typically cannot drive off the trail or road, which means you have to haul your rifle and gear by foot 150-yards from the road. Yes there are places that you can drive anywhere you want without restrictions.
Cannot shoot over or across any roads, trails, rivers, streams, or bodies of water.
No tracer or incendiary bullets, and no exploding targets.
Some Counties have restrictions on target shooting that are part of overall fire restrictions during certain dry months.
Pick up your trash, including brass, water bottles, cardboard, paper targets, etc. And don't be one of those ass-hats that haul old TVs or other appliances out to the desert, shoot them full of holes, and then leave them.
If you leave your steel targets out, no matter how remote you think they are, they will likely disappear. Ask me how I know.
Not sure about Utah, but in Colorado and Wyoming, it is your responsibility to know where you're at, and not cross (trespass) on to private land that may or my not be fenced or posted with signs. There are many private parcels inside or bordering public land, more than most realize. And not all the maps that are sold over the counter show them. If caught, you can be arrested, and fined/jailed (fines are quite high in Wyoming).