Target shooting is expressly permitted on National Forest Lands. However, finding a good place to do it is not necessarily straightforward. What would be good strategies to find a place to target shoot that has a good berm, is far enough from populated areas so as to not bother other people, and is still relatively close to a road to make carrying firearms, targets, and ammunition less of an ordeal?
As most of my experience is in the White Mountain National Forest in New Hampshire, this answer may be biased towards that location.
I typically try to find promising places within the national forest with satellite and topographic maps, then drive to the place to actually scout it out.
A promising place would be a clearing or a sandpit, at least a mile from any building. Swamps may appear to be clearings, but will be less than ideal for shooting due to the difficulty of setting up targets and the potential for ricochets off the water. Under powerlines can be a good place too, though more care must be taken.
Once at the location, find a good dirt or sand berm that will stop bullets. If there is not one, the site is not suitable. Pay attention to how recently people appear to have been there, and if there are empty casings or target fragments on the ground. These will be a good sign to see whether other people have shot there. There are many unofficial established shooting spots in national forests that are used by many people and not publicized.
If you find a good place, it is best not to spread the word, as too much exposure leads to places like these being overused, damaged, and shut down.