Oslo is an eco-friendly capital, and one of the most sustainable cities in the world. It is a compact city, so you can easily walk or cycle around town. The skiing slopes are just a few tram or metro stops away, and you have plenty of possibilities for outdoor activities all year around, such as: biking, cross country or downhill skiing, swimming, and sledging. The forests and hills surrounding Oslo, are often referred to as “Oslomarka” or “Marka”. Despite the fact that Oslomarka is frequently used as a hiking area by Oslo’s inhabitants, the wildlife is rich and varied, with animal species such as wolf, beaver, lynx, elk, and deer lurking around (but no polar bears unfortunately, you’ll have to travel to Svalbard for those).
The Oslo fjord islands are great to explore during summer, and there are several boats taking you out to sea for the same price as a tram ticket. In the winter (and all year long really), it is popular to visit the many saunas available, such as Oslo Kok, Salt or Oslo badstuforening, all situated on the sea so that you can go for a quick (and cold!) swim as well. If you like running, there are several public runs to attend, such as Oslo Marathon in September, and Holmenkollstafetten in May.
You can read more about green living in Oslo on Visit Norway and City of Oslo (Oslo kommune).
If you want to venture outside of the city and into the wild, make sure to check out The Norwegian Mountain Code by The Norwegian Trekking Association.