I think the best experiment of soccer in İstanbul can be gained by watching one of the "derbi" match between İstanbul teams -Galatasaray, Fenerbahçe and Beşiktaş-. Football is really important for people who lives in Turkey and these teams have really massive supperter groups all around the country.
2 days of every year, life literally stops in İstanbul. That days are the days of Galatasaray - Fenerbahçe matches. In last years this macht became the one of the most important and tense rivalry in European football. Too many people are waiting for it, talking about it, making jokes about it, even fighting about it (not dangerously :) So i thing that if you really want to experience the football in İstanbul, you must watch these macthes in that amazing atmosphere.
Update on Besiktas -Inonu stadium-
The stadium is about to be demolished (Summer 2013), so Besiktas will be playing its games either at Galatasaray's stadium or Fenerbahce's during 2013-2014 football season.
Galatasaray Turk Telekom Arena
Fenerbahce Sukru Saracoglu Stadium