The quickest way to get into the centre of Stockholm from the Arlanda Airport is to take the Stockholm-Arlanda Express. You will be at the Central Station in the middle of the city within just 20 minutes. If you find yourself a terminal 2, 3, or 4 you will have access to Arlanda Södra (south), but if you are in terminal 5, you will take Arlanda Norra (north). There are often discounts available, particularly at the weekend.
If you are heading for downtown Stockholm, the easiest way to get there is by using the Flygbussarna airport buses. They will also stop at other bus stops on route, and in the inner suburbs. If you are landing at the Arlanda Airport, these buses run at regular intervals, while those that make the journey between Bromma, Västerås and Skavsta airports connect with flight times.
A third alternative is to rent a car. This is easy to do and the majority of car rental companies have offices at all four airports. You can make your booking in advance or when you land. In either case you will need a valid driving license and a passport. Be sure to find out whether your car-hire price includes tolls because there are road tolls in inner city Stockholm for vehicles that are registered in Sweden.