If you only remember one thing about hiking with a dog, make it this: point the car at a national forest and you almost cannot go wrong.
Nearly every national forest in the country welcomes leashed dogs on its trails, which makes them the friendliest federal land there is for a dog owner. Here is the full verified list.
A 6-foot leash is required in developed areas like campgrounds and trailheads, and out on the trail your dog needs to be under control. Beyond that, the door is wide open.
There are more than a hundred of them, near almost every mountain town in the country. Find the one closest to you and go.
In developed areas, no, a 6-foot leash is required. On the open trail many forests allow voice control, but rules vary by district and a leash is always the safe choice around wildlife and other hikers.