The Necessities

Clothing:

Water shoes, sandals or canvas athletic shoes
Quick-dry pants, shorts, shirts
Quick-dry underwear
Fleece jacket and pants
Socks

Food/Water:

Water filter/purifier (Never drink unfiltered water from the stream!)
Water bottles/Water bladders
Energy bars/snack food
Freeze dried/dehydrated food
Backpacking cookstove
Fuel
Cookset

Sleeping:

Sleeping bag, preferably synthetic-filled
Sleeping pad
Ground sheet
Tent (Optional--know the weather)

Stuff:

Maps
Compass
Flashlight/headlamp
First aid kit
Walking staff
Inner tube/raft for floating packs across deep holes
Dry bags, to keep equipment dry and functional

Climbing/Rappelling equipment:

8mm to 10mm static rappelling rope
Tubular webbing
Light weight harness/Swiss seat
Carabiners
Rappel device (Sticht plate)