Transportation from Stockholm-Bromma Airport to your destination: 

Stockholm Public Transit:
SL Bus 110 runs to Alvik, where the metro train (Tunnelbana) or the streetcar stations are located;  and
SL Bus 152  runs to Alvsjo Station (Stockholm's International Fair) via Solna & Sundbybeg.

Airport coaches travel directly between the Airport and the City Terminal in about 20 minutes or so -

A new light railway (Tvairbanan) to the City
is 1 km from the terminal
(but outside of airport boundaries).  It connects to the Central railway station within 20 minutes.

Transfer to either Arlanda or Skavsta Airport:
(a) to Stockholm-Arlanda Airport. 
Take the Airport coach the City Terminal; and another coach - or high speed train (Arlanda Express) to Arlanda Airport.
(b) for Stockholm-Skavsta Airport transfer visit:  flygbussarna.se

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