Nobel peace centre. Approx £7 adult entry. Gives an insiteful history of norways rich peace price history. Worth a visit.
Oslo opera house. Beautiful architecture both inside and out. Definitely worth a visit as it is free entry! Walk onto the roof to get the best view of Oslo in the city.
Viking ship museum also on museum island.
Kon Tiki Museum. Best museum in Oslo I'd say! Get a fjords tour that departs every weekday morning from the city centre that drops you off in museum island. Get off and do the fran museum , kon tiki, national maritime museum and holocaust museum. The walk back to the city is about 45 minutes. It's very cultural as you pass through numerous green parks and niche norwigen houses.
Oslo royal palace. Worth a see. Right in the heart of the city and free to walk round the grounds.