Take a walk up through the tea plantations south-west of the town to the top of Single Tree Mountain for a splendid view over Nuwara Eliya. From the hill top you can walk north for a couple of kilometres along the ridgetop to a nearby village and get a tuc-tuc/three-wheeler or walk back to town from there.
For more sightseeing you can visit Victoria Park and Hakgala Gardens. Lake Gregory lies less than a mile from the centre of town and offers a splendid and peaceful environment for sightseeing and boating. Boats can be hired from the area.
You can also explore the tea plantations and factories and stay at one of the colonial bungalows near the plantations for easy access and special tours.
Playing golf is also an option, along with heading to the racecourses the only proper ones you'll find in Sri Lanka. These are both traditions left over from the colonial era.
Take a train trip from Nanu Oya towards Kandy. For the colonial experience book a 1st class ticket in the observation car. For an authentic local experience book a 3rd class ticket and hang out of the door of the carriage.