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I mean as a tourist you would probably want to first stroll Karl Johannes Gate (commonly known as Karl Johan) - that is the high street, the main road in Oslo - it begins at the central station and gradually ascends all the way up to the royal palace - it takes 45 mins or so to walk that stretch. There you find all the shops and some of the attractions too, including houses of parliament and the university of Oslo.

Within walkable distance from Karl Johan is the town hall square with the iconic (although dull looking) town hall - and next to it is Aker Brygge. I personally dont think Aker Brygge is very interesting, but people in Oslo seem to think it is a very nice place - it's where the boats come in, and it has a few upmarket restaurants. You will also find the Noble Peace centre in that area.

Oslo has a compact city centre - but Oslo is large in terms of size, since it contains much forested areas, surburbs etc.

There is also the Ski Jump , which was used during one of the Winter Olympics and if I am not wrong then there is a Winter Olympics Museum there - although that is not in the city centre - there are also a couple of museums , I havent personally been there but there is the Edvard Munch museum, and a viking museum or two.

In summer , one of the best places to visit is FROGNER PARK

Hi Julie,
         There is a travel site called tripadvisor dedicated for these information. Also look into hostelling international for booking stays in hostels which are cheap and better compared to some hotels.

Regards
Saturanga Kumar