bus or train cost from Birmingham airport to Cardiff

I don't know from buses, and the train fare will depend if you just walk up or get an advance ticket or a special. Go to http://www.nationalrail.co.uk/ and fill in the boxes in the form at the top. Use "BHI" for Birmingham International in the first box. It will give you the times, changing points (at least Birmingham New Street), platforms, prices and even maps of the stations along the way.