I think most of the major trips have been mentioned - just a few comments though.
County Pot, Wretched Rabbit and Mistral are the three entrances to Easegill which fit your description. County to WR is an excellent trip that we do fairly often with novices, the hard part is the navigation which often provides people with plenty of entertainment. In there own right County Pot is the more interesting, although WR has some interesting sections too if you're willing to explore.
Mistral to the Hall of the Mountain King, and then into the Cigalere streamway (which is brilliant) is also a great trip.
Link Pot - Mistral is also a good, though slightly squalid through trip. link Pot has a tight 14m entrance pitch though, which is easy going in, but can be a bit tiring on the way out.
Smeltmill Beck Cave which Dunc mentioned is well worth a visit, although personally I wouldn't bother going past the pretty bit as it gets bit boring!