Cardiff Castle, in the center of the city, is cross section of different time periods, from the Roman ruins, to the original Norman castle walls, restored and gaudily embellished by the Victorians in the 19th century (you can't miss the Clock Tower).
At the National Museum and Gallery, stop in to see the "Natural History in Wales" exhibit, dioramas of birds, animals and natural environments in Wales, and be sure to see all the Bronze Age gold and Celtic jewelry, stones and crosses.
The Welsh Industrial and Maritime Collections has a recreation of a dock at the turn of the century, ship models, and the story of shipping in Cardiff. If the weather is gray or rainy, step into the Indoor Market, a huge Victorian building that houses all kinds of stalls, a market in operation for generations.
On a nice day, take a run in Bute Park, once the grounds of Cardiff Castle or relax in Roath Park and Lake, boating on the lake and children's playground.
Outside Cardiff, visit the open air museum, Museum of Welsh Life at St. Fagans. The museum has over 30 authentic historical buildings, representing life and work of the Welsh people from the Middle Age to the present. St. Fagan's Castle is the biggest of them on, a 16th century mansion. Castell Coch (Tongwynlais) is a fairy tale castle of the 3rd Marquis of Bute, built in the 19th century on the remains of a much older castle.