Slate mines: plenty of free to enter around Blaenau Ffestiniog - but you need to know where / what you're doing. Examples are Conclog and Wrysgan. Or, pay for a trip around Cwmorthin led by GoBelow - Cwm is awesome and you'd be in there for a minimum of c.5 hrs for the round trip.
Metal mines: again, plenty but for those in the Conwy Valley - eg. Parc, you really need a guide and for free access, need to register, foc, with Cave Access Ltd.
On Anglesey, we have the Parys and Mona mines on Parys Mountain near Amlwch. Parys Underground Group meet on Wednesday evenings and if you hold BCA Green Card Insurance, we'll take you in for free otherwise you will need to join PUG as a temporary member (£10 - valid for 17 days) for your insurance cover but I can promise you an amazing 2 - 3 hr trip.
The lead mines of Halkyn Mountain, Flintshire are a little more problematical and you'd need to contact the local group, UCET who would probably take you on one of their forrays.
There's the Thursday Nighters who go out, strangely enough, on Thursday evenings and go around a variety of mines - usually in NW Wales - you could tag along, a friendly bunch. Look for their latest trip advertised in the forum and send a pm to one of the leaders - who will usually be the person that posts...
There, that's a few ideas without knowing your experience level...