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:
- From Frognerseteren to Tryvannstua, Skjennungstua and Ullevålseter
- From Sørkedalen to Kobberhaughytta
- From Sognsvann to Ullevålseter
- From Skar to Øyungen
- From Tverrsjøen to Svarttjernshøgda.
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".