The best introduction to Derbys. caving have always been Carleswark and Gautries, enough to play and more if you want it.
If you want wet and using tackle, go for P8 or Giants.
If you want vertical go for Eldon, Rowter, Knotlow or Water Icicle.
If you want really nice cosy caves, go for Waterfall Hole (but watch out for wetness), Cumberland/Wapping, or Jug Holes (don't forget your spanner!)