Arrival by air to Arlanda:
The main airport Arlanda is situated approx 40 km from the city centre. It is recommended to take public transport from Arlanda to the city centre and to continue with a taxi or underground from there.
The airport buses Flybussarna are the cheapest, but slower (45-50 minutes). You buy the ticket either in advance over internet (e-ticket), inside the terminal or from the driver. The bus ride takes 45-50 minutes. Stay to the very last stop Cityterminalen. 
The Arlanda Express train is more expensive but very fast (20 minutes), even faster than a taxi! Buy the ticket in the vending machine at the Arlanda Express terminal with cash or credit card. 
From Arlanda Express terminal in Stockholm you can continue either by taxi or by underground. 

Arrival by train:
The train is very convenient for local arrivals as it stops in the middle of the city center, and the subway is easily accessible from the train station.

Continuing by underground:
Go to the main railway station Centralen. Go downstairs, turn left and follow the long passage to the underground station T-Centralen and take line 14 to the stop Stadion.
You can buy tickets for the local "Stockholm Transport" system (SL) from ticket offices and vending machines, but not on board.