There is one train an hour from Rhoose station (about 5-10mins walking time from the Flying Club on the Southside of Cardiff Airport) to Cardiff Central (the journey takes about 20-25mins) & one train an hour in the opposite direction. A timetable should be easy to find online.
You could then walk to Ninian Park (its not that far) or catch another train.
Hope this is of use
Fried Chicken