Depends where you are in South West Hamburg! If you are south of the Elbe then you should see if there is an S-Bahn or train line near you to get to Hauptbahnhof, then jump on the U-Bahn to St.Pauli or Feldstrasse stations that are right beside the Fanfest. Also south of the Elbe you can get a bus to Finkenwerder, then take the ferry to Landungsbruecken and either walk or take the U-Bahn.
Best bet is to look at H.V.V. website www.hvv.de (it is also in English) go to "timetables" enter your street name as departure point and "Feldstrasse" as destination then it will give you the best routes.
Don't know much about smaller galleries - there are obviously the Kunst Halle etc. as major museums and I know around Jungfernsteig there are art shops in several of the malls selling original paintings but I'm not really into seeking out art.