If you're not ready to make any big investments, I would suggest a cheap tripod and some decent flashlights to do light painting. You might also want to look into a shutter remote because they're cheap, allow for longer exposures (which may be necessary if your camera isn't capable of a high, quality ISO level) and can help with camera shake (sometimes, even on a tripod, you move the camera a bit just by hitting the button).
Like ForgottenRails said, use your flashlight to shine on your subject to get good focus where you want. Then turn your camera to manual focus so it doesn't try to readjust. Do a long exposure and some light painting. You'll have to do some experimenting to figure out how long the exposure should be and how much light to paint in.
Once you figure out how to do that successfully you should get decent exposure. Here's a few of my personal results with that technique:
Reflections by
Gina, on Flickr
^Underground with no outside light
Into The Light by
Gina, on Flickr
^In an above ground tunnel with very little light
If you want to get more complex with it, you might want to also consider bringing along multiple light sources to light up different areas to give your pictures a bit more depth.
Maze by
Gina, on Flickr
In this one I used 3 different placed lights which are pretty apparent. I also used an off-camera flash to lighten the tunnel on the ceiling, and a fourth light to light paint the foreground.
When you're doing the painting you should be kind of random about it. Move the light constantly to avoid hot spots and try to light up the entire scene. Keep in mind that things that are closer to you will reflect the light stronger and therefore turn out brighter, so focus a bit more of the parts that are farther away.
As far as dampness goes, I've never used anything on my camera to protect it from water, but it might be a good idea to find a drybag that will fit your camera case. Usually when I go caving I'll put my camera in my padded camera bag, then put that inside a drybag, and then put that inside a drybag backpack. Just be careful when you take it out.
Hope that helps at all. Best of luck to you. Looking forward to seeing your pictures.