There are many places that can still be accessed for little or no money to hike and explore. The Blue Ridge Parkway for an example has many hiking trails to points of interest. Add a picnic to the equation and you've got memories that will last a lifetime.
You don't need to buy expensive equipment to start birdwatching. Each family member can track and identify birds seen in your own backyard or on wilderness adventures.
It doesn't cost anything to window shop at farmer's markets and local fairs. Many have free entertainment as well.
Many museums operate on a donations only basis...
Car enthusiasts love showing off their rides and most are free to the public.
Some state parks will give huge discounts to area residents on specific days...often free, too.