Looking for a town accessible by public transportation

Wrexham?

Cardiff?

Aberystwyth?

Bangor?

I don't know Wales well enough to help, but it might be useful for other responders to know whether you're thinking about northern Wales or southern Wales--or central Wales, for that matter.

We love the Conwy Valley in North Wales as it is easily accessible by train (to both Llandudno Junction or down the valley line towards Betws-y-Coed), you can get the bus around also, and you can easily spend 4-5 days walking and enjoying the scenery, and seeing interesting sights (Conwy for the castle, Llandudno on the coast etc). You can get further into Snowdonia using the Sherpa buses.

Betws-y-Coed has lots of accommodation and you can reach it by train (it is very popular though), and Llanrwst is a nice little town nearby, but we like Trefriw just to the north of Betws (you can walk to Trefriw from Llanrwst which has a railway station).

It is a cute but quiet little village, the woollen mill is really interesting, there are some lovely places to stay there, some great places to eat so you don't need to travel / drive in the evening, and the walks into the hills and forests from the village are fantastic!

Hope this helps.

Another vote for Betws-Y-Coed - which can be reached by train by changing from the N Wales mainline at Llandudno Junction.

The town is also served by Sherpa Bus that will take into the mountains for hiking.

An alternative to Betws y Coed would be either Conwy (walled medieval town with a ruined castle) or Llandudno, a Victorian seaside town with pier, and traditonal Punch and Judy show. Both have good rail and bus connections to the surrunding areas and the potential for some good walking - Great Orme, Conwy Mountain as well as walks in the Conwy valley.