Exploring Oslo by bike is a great way to see the city. From the Akershus Fortress, stopping at Snøhetta Architects’ fabulous Opera House (and be sure to take a walk on the roof), then through the heart of the city to Vigeland Park and the beautiful residential neighborhoods that adjoin it, there are well marked bike paths and friendly locals to help out if necessary. I can recommend Viking Biking (love the name) with its excellent multi-lingual guides and a variety of group and private tours. They will also rent bikes to more intrepid travelers who prefer to go it alone.
The Bygdøy peninsula is an easy bike ride from central Oslo and a great place to spend an afternoon or even a full day. Bygdøy is home to The Viking Museum and the Fram Polar Ship Museum — both are absolutely fascinating — as well as a host of others. The Bygdø Royal Manor and Farm, in continuous use since the Middle Ages, is on the peninsula and there are lovely forests and beaches with excellent picnic spots. It is possible to bike there and then take the ferry back across the harbor.
See also: Awesome Oslo – part 1