The currency in Norway is Norwegian Crowns (NOK and not EURO). You can use your credit card almost everywhere for payments (sometimes you might be asked for an identification with passport or id card) and for cash withdrawal (called Minibank).
The airport for Oslo is called Oslo Airport Gardermoen (OSL). There are two other airports which are sometimes also called Oslo, namely Rygge (RYG) and Torp (TRF) (see below), but they are much further away from Oslo than the Gardermoen airport. They are used by low-cost carriers, but they are only recommended if you are very short on money because of the long travelling times to and from the airport.
When you left the customs area at OSL airport, you should turn right and walk down to the end of the hall until you reach the stairs to the Flytoget (airport express train). The train takes you to Oslo Sentralstasjon (sometimes called Oslo S) in about 25 minutes. One ride costs 170 NOK. You are obliged to buy a ticket before boarding these trains — it is not possible to buy tickets on board. Tickets may be bought by swiping a credit card, at the distribution machines or at the NSB counters.
From Torp, the company Torpekspressen offers buses to the Oslo Central bus station (Oslo Bussterminalen) which is in walking distance (5 minutes) from the central train station. The ride takes about 1:45h and cost about 200 NOK for a one-way ticket and 340 NOK for a return ticket.
NSB trains to Oslo/Nationaltheatret run every 1 to 2 hours. The ride takes about 1:45h and costs around 240 NOK (plus 40 NOK fee if you buy your ticket on board of the train). There is a free shuttle bus connecting to the nearest train station, see here.
Rygge airport also has a train connection via a free shuttle bus, see here. A ride to Oslo takes about 50 minutes and costs around 150 NOK (plus 40 NOK fee if you buy your ticket on board of the train).
In case you would like to take the opportunity of coming to Oslo for a visit to other parts of Norway, we have collected a few hints and links: