Here's some ideas:
Run through a meadow
Hug a tree
Smell a flower
Hike a trail through the woods
Stop at a stream for a picnic
Swim in a stream or lake
Climb a mountain (or a mountain trail)
Explore a cave
See birds or wildlife
Mountain Bike
Photograph wild flowers
Collect leaves or pine cones
Enjoy the beauty and serenity of nature
Lie back and look at clouds
Find a waterfall
Take a canoe, kayak, raft or tube trip on a river or stream
Join a group and help maintain trails
Search for abandoned buildings/towns/mines
Check the park/ranger office for any scheduled activities
Go horseback riding
Fly a kite
Swing from a vine
Go fishing (with a license)
Skim stones across a pond
Catch a frog and let him go
Catch a (non-poisonous) snake and let him go
Practice using a map and a compass
Take lots of pictures
Run away from a bear
Don't feed that big kitty
Don't go near deer, (Bambi can kick the cr@p out of you)
Ditto for most other wild animals
Don't eat the mushrooms
Look for Native American artifacts, structures and paintings
Look for Civil War artifacts
Pick and eat your own berries, apples, corn, etc (only where permitted)
If it rains and you're stuck inside:
Card games (Solitaire if alone)
Checkers
Backgammon
Chess
Dominos
Mahjong (Chinese tiles game)
Go (also called Weiqi or Baduk), Chinese strategy game of white & black "stones"
Other travel- or full-size board games
Hemp or other string to make braided bracelets (friendship bracelets)
Supplies to make bead jewelry (beads, string)
Supplies to make native american-style jewelry/items (beads, feathers, string, wire)
Pad/paper, pens, pencils, charcoals and/or markers for sketching and drawing
Scrap material and sewing supplies to work on a patchwork quilt
Modeling Clay / Play Dough diorama
Popsicle sticks and white glue to make popsicle stick sculptures
Legos! (just don't leave them lying around to be stepped-on, it hurts!)
Pick-up Sticks
Barrel of Monkeys
Marbles (another thing to not leave lying around)
A good book to read
Rain Jacket, pants and boots to go outside anyway!
Most important
Be safe
Leave nothing but footprints (pack out all trash)
Take nothing but photographs (a few leaves, pine cones or berries are okay)
Have a good time