Erm.. as for mineral I'm not sure...
For slate.. definatly Hodge Close.. it's very easy to access, and has several safe, interesting levels and chambers worth looking at. You may alsol like old man quarries on the coniston old man.. though this involvs ascending a mountain to reach them but...
both are on this website: just search Old Man orHodge Close
There are not by any means the only ones, though many of the other ones such as Honister are still being worked. You might find Rigghead interesting.. that's on this site too.
Kein geneis kanaf - Cain gnais canaf
Byt vndyd mwyhaf - byth onddyth moyav
Lliaws a bwyllaf - LÃows o boylav
Ac a bryderaf - ac o boryddarav
Kyfarchaf y veird byt - covarcav yr vairth
Pryt nam dyweid - poryth na'm dowaith
Py gynheil y byt - Pa gonail y byth
Na syrch yn eissywyt - na soroc yn eishoyth
Neur byt bei syrchei - nour byth bai sorochai