I have done a bit of climbing with a DSLR, and a lot of hiking with one.

If you're climbing with it, you don't want it on your chest, and if you want it accessible, you don't want it on your back. This means that most of your options are holster style bags. I have a tamrac one which happens to be shower proof when closed (in reality it's been more than showerproof) but find one which suits your camera. DSLRs tend to go in them lens down.

If you're wearing a climbing harness you can't easily attach a holster to a belt loop, so consider wearing a lightweight waist belt with the holster on your side. Place the holster on the side of your non dominant hand, so that you still have access to any gear attached to your harness (you will have to sacrifice the usage of gear loops on the camera side).

To avoid dropping it, you can use something like a BlackRapid. But if you want to minimise your spend, take a length of paracord (or very thin rope), attach one end to the centre of the back of your harness, pass it over the shoulder opposite the holster, and then attach it to the camera in the holster. If you drop the camera, it will probably just hang in front of you, but worst case it will swing below you, and you can pull it back up.

When Hiking, I use the same holster, but I attach it to the waist strap of my rucksack, this puts it in a convenient place for access, and provides a nice arm-rest for my rucksack.

If you regularly want to take your camera to interesting places - get some decent accidental cover insurance. This can put your mind at rest very easily - it certainly makes me feel much more comfortable with my camera in dangerous locations.