Taplas is a very good guide. They also organise frequent concerts in the Norwegian Church.
Folk Clubs as we know and love them are thin on the ground in the city centre. The "English" trad scene never really took a hold here. Howeveer, there has been a long history of blues, contemporary folk (for want of a better phrase) and singer-songwriter stuff.
There is Roath Club, which is actually now in the Heath area, St joseph's Club on wednesdays - not much trad (but it's always welcome), plenty of Americana. There's the Blue Dragon Hotel, Newport Rd on Fridays, which is basically Tony Breen's gig with an open mike session. He does mainly Irish and American stuff (folk & country) and it's not a totally attentive audience, but it's a friendly, sociable place.
Session-wise, as mentioned, the Conway, Pontcanna is a general tunes sesh on Thursday nights, and the Mochyn Ddu on Mondays is a Welsh music sesh.
If you don't mind travelling a bit, there's a fair bit out of town. Llantrisant on Weds and Newport on Thurs. And Monmouth, Swansea, Pontardawe and Gower further afield.
There is also a strong "student" acoustic scene in bars such as Journeys, Clifton St, The Toucan, St Mary St, Hoffi Coffi, Woodville Rd.
Also try the Folkwales.org web pages for general info, though only Llantrisant club is listed.
Do get in touch when you get here, all the best.