Hmm...
Mendip has a lot of good caves that are quite do-able by relatively new cavers even if the guide book does say "VDC".
Goatchurch, IMO is now a difficult cave because all the climbs ars so slippery. I have never done the final series, the squeeze looks and sounds ghastly.
Rod's Pot. Good cave.
Swildon's - get through Sump One and visit Sump Two! Or look at the Black Hole. You can even tackle the Short Round Trip suitably attired as long as you have someone with you who knows it, but it does rather depend on whether the Mud Sump and Troubles are passable. Don't be put off by names like "Greasy Chimney", it's an easy climb.
GB needs a key but is another must-do. The climb down the rocks in Main Chamber is slightly awkward for beginners and I like to use a handline - more for going down than coming up. You are then in some BIG cave! The extension to Bat Passage is optional but worth the effort. It's not really hard, just needs standing on someone's back to get a short ladder onto the hanging chain and then some ingenuity to get to the top of the chain!
Eastwater. Another must. If you are short of tackle, the Thirteen Pots is a good trip. Take a short rope just in case. With tackle you can do the Twin Verts or the Dolphin Pot route and get into some serious bits of cave. Nothing pretty until you go deep down to Regent Street (which is a bit of a slog) but it's an awe-inspiring cave.
Charterhouse, Reservoir Hole and Upper Flood need leaders but are beautiful caves.
Likewise, get a leader from the BEC to take you round St Cuthbert's. It's a tough cave and has some amazing decorations, you will enjoy it. You may even get a sporty trip crawling under the Beehive and trecking down to Sump Two, depending on the weather.
Yup, there's plenty under Mendip, some places are better known than others. Hope my memory is Ok.