If you fly into Cardiff during the day, there is a couple of public bus services which stop outside the terminal building and carry on into the city centre. The journey is about 45 minutes, and cost about £3. You'll need exact change (which could be a real pain if you've only got big notes on you)
I was staying in London and used my rail pass to take a day trip to Cardiff. It's two hours from London's Paddington station and you can have the whole day there before heading back to London. Buses also go from London and take just over 3 hours.
Here's the Rail Europe site: Rail Europe
For direct link to BritRail timetable click here: BritRail
Take a trip on the road train linking Cardiff Bay with the Cardiff Bay Barrage through the heart of Cardiff Docklands.
Cardiff has a widespread bus network - you probably won't need unless travelling outside town as the city is fairly compact.