Hi Ray,
Due to the local geology in Mendip, most caves do not have many pitches, so personally I'd drop this criteria if you want to experience what the area's really like.
My personal recommendation for a weekend in Mendip would be to stay at the Wessex Cave Club. You can walk from here to Swildon's Hole, which is one of the best sporting caves in the country; one 20 foot ladder (borrowed from the Wessex) will give you options on some fantastic trips. You can also walk from here to Eastwater, which is another Mendip classic and is a must do, for anyone who hasn't experienced the 45 degree dip limestone; an evening trip to the 13 pots would be a great introduction to the cave, involving 40 feet of ladder from memory.
My final recommendation for the weekend would be GB cave which is a short drive from the Wessex. A trip to Great Chamber (involving one short ladder climb) and exit via the Devil's Elbow is a brilliant and varied trip.
If you are set on doing pitches, you could go to Thrupe Lane, Hunters Hole, or Rhino Rift, but you're better saving the SRT for other regions.