Rent prices in Norway, like almost everywhere in Europe, depend mostly on location. A three bedroom apartment in Oslo could cost you 15,000 NOK however a three bedroom house in Larvik, Vestfold could be just 10,000 NOK per calendar month. The rule of thumb generally is further you are from town the less expensive the rent gets.
In Norway a deposit is usually required when renting. Deposit prices may vary but the norm is three months rent. It is also common to pay your rent a month in advance so you'll need this money to start.
Sometimes you will be able to use a guarantee, a contract that stipulates any damages incurred will be covered by you but many places no longer accept these contracts.
As a renter in Norway you will not be expected to pay council tax on top of your rent, this is something the home owner must pay for.
Many rental accommodations do not allow pets, though there are exceptions. ‘Dyrehold tillatt’ is the Norwegian term for pets allowed, on the popular sites used for searching for leasings (Finn.no) there is a check box you may tick for this option. You may have to pay an addition deposit or your leasing contract could likely include that you will have to pay for all pet related damages.
If you can’t find an apartment suitable for you on a search you can also post an advertisement on Finn.no telling a little bit about yourself and what you would like and the price range you would like to pay, be warned that you will probably need some Norwegian knowledge to do this as Finn is a Norwegian site.
Long term tenants are usually preferred and it is normal to have to give three months notice when moving out.