It depends on what sort of shooting club you're after.

Quarry Hunters meet near Blackwood on Sundays and shoot HFT courses and informal plinking.
First visit is free.
For me it's a 45mile trip but well worth it.
Keen HFT'ers from all over travel there.
It is after all one of the homes of UKAHFT.

Quarry Rifles meet on Saturdays and they're an FT club.

Swansea Rifle Club covers has indoor and outdoor ranges (outdoor range open Monday eve and Saturday am).
They're predominantly an FAC club but there are air rifle and air pistol sections as well.
They are looking to start up an FT or HFT section. They haven't decided yet.
Good place to learn about using and loading firearms.
Indoor 25m range excellent for setting up guns in bad weather.

There are gun clubs of various kinds all over South Wales though they're not all easy to find at first.

As WILBA says there is the Tondu club near "Knees", they shoot mainly FT.