A nice little mine/ cave system I use for first trip novices is the Wapping Mine & Cumberland Cavern system at Matlock Bath. The system is quite dry (very few wet patches) and no special equipment is required. It is one of those systems where you do need to go down and get to know it with someone who does know it. It is a real labyrinth once you get down the bottom of the main mine workings.
Only disapointment on this trip is the vandalism to Cumberland Cavern system (old show caverns).
For those not been there, Cumberland Cavern entrance is sealed and you must enter through the Wapping Mine end (south west from the New Bath Hotel). If you want a visit sometime, or decent directions to entrance, drop me a line.