Plenty around Blaenau Ffestiniog! Have a look at the Cwmorthin pages on here. Croesor and Rhosydd are probably a bit extreme for you to look in until you get more experienced. Fron Boeth in the Croesor Valley is worth a look in. Above Cwmorthin is Conclog - if you don't mind getting wet wading through the entrance adit. Opposite Cwmorthin is Wrysgan with simply HUGE chambers.
If you're willing to pay a bit more, have a look at Go-Below - Miles takes parties through Rhiwbach and also Cwmorthin.
There's the show mine at Llechwedd but a bit further down in mid Wales is one were you pay but will be led into another unsanitised slate mine - Braich Goch - follow the Corris Mine Explorers ad on this website.
My favourite is Cwmorthin. Ideally you'd need a guide as there's miles of tunnels and chambers spread over many floors...
Basically, just look around this website - use the search for slate mines in N Wales then look around any that have a photo gallery and take your pick!
Suggest you post on here before you visit any to get up to date access and safety info.
Clive