I’m from Norway and been living in Oslo the last seven or eight years, and can recommend a couple of cool places to check out.
Lufthavna MIR is a cool underground concert venue. A lot of free form improvisation, experimental and noise music stuff. Kafé Hærverk is a pretty similar venue in terms of music / scene. Blå is another venue down by a river called Akerselva - also a lot of cool shows both there and a nearby venue called Ingensteds. Be sure to check out the program at Parkteateret too. Parkteateret is located in one of the more trendy parts of town called Grünerløkka. A lot of nice cafés and bars there.
The best record ships is Big Dipper and Tiger. For music equipment go to Filter Musikk. The prices are pretty steep here though.
For food and drinks I can recommend Isakaya which is a very cosy Japan styled bar. Kamai for some really good asian food. Vippa is basically a hangar at the docks with a lot of food stands with all kinds of food. The weather has been great the past month, so if this continues Vippa is a fantastic place to go and to sit outside watching as boats and ships go by.
I would also recommend to take the ferry out to the islands in Oslofjorden. A pretty short ride and inexpenssive ride. I would especially recommend Gressholmen where you can buy food and drinks at a little restaurant - or you could just bring your own and have a picnic.
Astrup Fearnley is a nice museum for contemporary / modern art. There’s a pretty cool sculpture park called Vigelandsparken, you should check that out.