I have not been there myself yet (prolly need to get my PPL first!
)... But word on the street is to choose Barkarby, ESKB, rather than Bromma, the latter being mostly a quite expensive airliner and bizjet place these days... But that is something you'd have to look into.
Perhaps you might find any of these interesting:
AIC for foreign visitors:
http://www.lfv.se/upload/ANS/AIP/AIC%20A/A3-07.pdf
FPC, for weather, NOTAMS, AIP and such:
http://www.lfv.se/templates/LFV_LFIStart____4353.aspx
A few helpful hints about light GA flying in Sweden (if you gloss over the first two paragraphs about our club!):
http://www.aeroklubben.se/International/English.aspx
From what I hear about some other European countries, Sweden appears to me to be a quite GA friendly place... I hope you'll feel welcome here!