Upvote:3
As the others have said, the most dramatic nature in Norway is on the west coast, with the fjords and many of the famous hiking trails. That is most likely what you have seen in pictures and videos of Norway.
But the beauty of Oslo is that you can take a metro to the forest and hike as far as you want. I think it's pretty cool to do as a traveller because it's an authentic local experience. On the weekends in summer (Sundays in particular) it's very common for Norwegians to go for a hike. It's also free which is a nice bonus in Norway.
Outtt is a good traveller-focused site for more information on hiking in Oslo and some suggested routes, including some of those mentioned by Stewie. Some of my personal favourites:
Upvote:5
If you want to go trekking close to Oslo, I recommend Nordmarka, often just called Marka.
Have a look at the official website here, for some information. There are many good starting points in Marka, including Holmenkollen, Frognerseteren, Sørkedalen, and Sognsvann. Many of which are easily accessible by tram or bus.
Once in Marka, you can stay at some of the DNT cabins, such as Kobberhaughytta, Tømtehyttene, Smedmyrkoia, Katnosdammen, Sinnerdammen.
Some of the most popular trekking routes are:
Having said that: The west coast of Norway (and Lofoten) offers the most spectacular views, and it's where you'll find what most would consider "awesome Norwegian nature".