The city of Oslo is the capital of Norway. Here you will find a history city dating back more than a thousand years. Today, Oslo is a bustling European metropolitan area that offers everything to keep a Norway visitor happy, fed, and entertained. You could spend weeks in Oslo without doing the same thing twice!
Bergen is one of Norway's most popular destinations for good reason, since Bergen isn't far away from Oslo, has a very scenic location on the west coast of Norway, and is one of the warmest cities in Norway in winter.
Trondheim was the first capital of Norway. It is a smaller city than the first two picks, and the quiet historical districts remind of old fairytales. Maybe it's this kind of atmosphere that makes it so special to visitors that want to take things slower and relax a bit more. Trondheim is especially beautiful in late summer and fall. Make sure to enjoy the fresh seafood available here and visit Nidaros Cathedral, the largest church in Northern Europe.
Lillehammer is a city for the active traveler. Just remember that the city of Lillehammer doesn't only offer skiing! In the summer, there's fishing and hiking, the Maihaugen open-air museum, a waterpark and a nice pedestrian area in the center of Lillehammer. In the winter, of course, this Norwegian city outdoes every other place, with winter sports, sleigh rides, a winterized amusement park, luge track, ski jump, ice fishing, etc.
Yes, it's a rather tourist-y destination, but for a nature loving traveler the beauty of the Geirangerfjord surroundings easily trump that. If you can't stay long, consider Geiranger as a 1-2 day trip from one of the bigger cities in Norway. This is one of Norway's must-see destinations.