River Rafting 
Swimsuit; tennis shoes or Teva-like sandals with secure ankle straps; t-shirt (quick drying material is best); easy-dry shorts or running tights; safety strap if wearing glasses; waterproof camera; plastic bags for anything you want to stay dry; dry clothes for after the trip; and lots of adrenaline.
Mountain Climbing
Below are supplies you will need on a two day non-technical climb up mountains such as Cotopaxi and Chimborazo. For the more technical climbs, more specialized equipment is needed. For nearly all tours, the items listed under "Equipment" are provided by the tour company. All equipment may be rented in Quito as well. Helmets are the exception to both of these statements, they are rarely provided and are also very difficult to rent. 
Equipment
Mountaineering boots; crampons; ice axe; carabineers; climbing ropes; harness; and gaiters. 
Personal 
Food; water; headlamp; 3 sets of batteries; Glacier Glasses; lip balm; sunscreen; knife; 2 water bottles; Two thick insulating layer tops; one thin, one thick; insulating layer bottoms; Gore-Tex type hooded coat; Gore-Tex type bottom; glove liners; Gore-Tex Gloves; 2 pairs of socks; warm hat; camera - keep it in your inside pocket, it'll freeze; extra change of clothes; and shoes for around camp.