As an Oslo native, I would recommend you to stay off the central station/lower Karl Johan area, not because it's dangerous but it's simply not very interesting and yes, it's less pleasant at night. I never hang out in this part of the centre, except when travelling in and out - and to visit the opera (to walk on the roof and sit in the sun).
In addition to the Viking Ships, I personally like Folkemuseet in the same area (Bygdøy) on a sunny day (old buildings, folklore, see how apartments were furnished in 1920s etc). In summer you're able to take a boat to Bygdøy (to the Kon - Tiki - Museum)
In summer, a boatride out to the islands, for a picnic is a must - at least for people living in Oslo. The metro/bus pass is valid, and you could even go "island hopping"
If you go to the Vigelandspark, and it starts raining or something, you could visit the city museum- Bymuseet which is right there and is free and somehow cute.
There's also a "Pop Centre" http://www.popsenteret.no/ArticleDetails.aspx?ArtId=111&mid=226 which currently features an exibition of Norwegian metal...
In terms of neighbourhoods, I enjoy strolling around the more multicultural Grønland in addition to Grünerløkka; the former has cheaper food and drinks and the latter has all the designer shops and coffee shops etc.
You could visit the cemetary where national poets etc are buried, "Vår frelsers gravlund", and walk through the old quirky back street Damstredet with colorful wooden houses.
I slso recommend walking along the river - Akerselva, like us locals, e.g. up from Grønland right beside the metro stop - and up to the neighbourhood Sagene - the former industrial centre.
For modern art, in addition to Astrup Fernley, there's the Museum for Samtidiskunst, and Henie Onstad gallery outside of the city center. Also, for art and cultural events, you could check www.underskog.no</strong> - the daily calendar offers some of what's on the more or less underground scene, and many things free.
Nightlife is (yes expensive) good around Youngstorget (Mono, Internasjonalen, Bonanza, Hells Kitchen, Tilt etc, in addition to concert venues such as Rockefeller and Sentrum Scene.) and Grünerløkka (Ryes, Parkteateret, Schouskjelleren, Bar Boca, Blå - fun and free concerts on Sundays etc) Grønland, - and more upscale around Bogstadveien/ Solli plass. In summer, bbq and drinking in the parks is equally fun - and cheaper.