Bakklandet is a preserved, old town portion of Trondheim directly across the river from Nidaros Cathedral. This areas has many coffee shops, restaurants, and great views of the Nidelva River.
You can hike to the top of one of the mountains in Trondheim to Estenstadhytta, which is a small cabin with food and beverages. It's a very popular place amongst locals and there is wonderful lakes and forests along the way up.
https://estenstadhytta.no/
If you're looking for a spot to relax by the fjord, go swimming, or you just want to see where Ole proposed, Korsvika is a really great option. You can bring food and barbeque on the beach.
The Fortress is an interesting historical sight where lots of cultural events are held in Trondheim. It's a beautiful place to take a walk and see views of the city.
If you're looking for a walk along the Trondheim fjord, Ladestein is a very popular trail that you can follow. It has great views with various small restaurants that you can stop at along the way.
Munkholmen is an island in the Trondheimsfjord, approx. 2 kilometers straight outside Trondheim. The island has a rich history, and has been used as a court, monastery, fortress, prison and defense facility. During September, a boat goes every hour from 11:00-3:00PM on Saturday and Sundays and tours are offered in English.
Nidarosdomen is by far the most visited sight in Trondheim. It is the oldest cathedral in Northern Europe. You can go into Nidarosdomen and see the architecture of the church. Directly next door is the Archbishop's Palace, where the Catholic Archbishop lived when the church was Catholic. It also houses lots of historical items such as crowns from king and queens, old books, etc.
The Old City Bridge is a very famous spot to take photos! It has a great view of Nidaros Cathedral, the Nidelva River, and Bakklandet with picturesque, colored houses on the river.
Open air museum with more than 80 buildings from Trondheim and the Trondelag area. Beautiful surroundings, and from the ruins of King Sverre's Castle you have a wonderful view over the city and the fjord. The impressive main exhibition Livsbilder, Images of Life, in the public building reflects childhood, youth, adulthood and age in the region during the past century. The rural section shows the development in Trondelag buidling styles from the fjord to the mountains, and how people lived in the 18th and 19th centuries in the different country areas.
The Tyholt Tower is a large tower in the middle of Trondheim that has Egon Restaurant at the top. As you sit and eat here, the restaurant turns and has 360 degree windows, so you get really amazing views on Trondheim and the fjord. They have a pretty affordable pizza buffet during the lunch time!