It is best to do some preparation yourself before visiting. Get familiar with the history and geography of the year.
Known as the Galata Mevlevi Museum, this is an old Sufi Dervish lodge founded in the 15th century. It includes a complex with courtyard, cemetery, and the monastery, now converted into a museum of Sufism. This is one of the only places in Turkey where you can learn about the monastic order, with clothing and artifacts of the Sufi order on display. The most interesting sight is the whirling Dervish ceremony that is sometimes held here.
If you're in Istanbul, you've got to see the whirling dervishes spin into ecstatic ecstasy with Allah. The music they dance to is played live and is almost as mesmerizing as the dervishes themselves. Non-Muslims are most welcome to the sema (the ceremony) because of the fact that this particular sect of Islam believes that all are equal in the eyes of God. Cameras and video cameras are ok even, just turn off your flash so you don't blind one of 'em. The hall where they get their spin on is actually part of the train station on Galipdede Caddesi just south of Tunel Square, at the southern end of Beyoglu's Istiklal Caddesi.