I ALWAYS have a sidearm with me while camping.
Here are a few regulations. Check which forest you will be in as there might be additional ones. Have fun and stay safe.
Federal Rules
According to the Code of Federal Regulations (title 36), recreational shooting can take place on the National Forests and Grasslands under these conditions:
You are at least 150 yards from a residence, building, campsite, developed recreation area or occupied area.
You are not shooting across or on a National Forest System Road or an adjacent body of water.
You are not shooting into or within a cave.
You are not shooting in any manner or place where any person, property or resource is exposed to injury or damage as a result of such discharge.
You are not firing any tracer bullet or incendiary ammunition.
Targets and Alcohol
On the Arapaho and Roosevelt National Forests and Pawnee National Grassland :
You may only shoot targets that are:
cardboard,
paper,
self-healing,
manufactured metallic (metal targets specifically designed for firearms),
or manufactured thrown-type clay (clay pigeons)
Wooden backing on targets is not allowed
You may not possess an open container of an alcoholic beverage while discharging a firearm. (Order 10-00-2011-01)
You may not use an explosive, including exploding targets. (Regional Order R02-2013-01)
ETA: One of our rules (sayings) is "Suns out, Guns out". We put 'em up at night.