Also Lake Taupo should deserve yourtime - probably the most beautiful lake in North Island. Nearby there is a nice trail along Wairaki River, and see Huka waterfall. You can also do the exciting and famous "Shotover" jet boat ride. I believe the Jet Boat ride originated from Huka near Lake Taupo.
Hamilton has not much to see.
Be sure to visit the Te Papa Museum at Wellington.