The cheapest but relatively slow means of transport is a bus. On the European side, the main bus lines start from Taksim Square, Besiktas and Eminonu. On the Asian side the departures are mainly from Kadikoy, Uskudar and Bostanci. The destination of buses is written on the sign on the front of the bus and the route is written on the sign next to the entrance door. These buses go to Rumeli Hisarustu and your destination is one stop before the end of the line.
Buses are frequently crowded.
No smoking is allowed. People exit from the rear. Taxis are plentiful in Istanbul and are inexpensive by US standards. The cost is what the meter says. Going to the university campus is accomplished by asking for Etiler or Bebek and then Bogazici University.