The cheapest and best way to see Istanbul is on a commuter ferry. The Şehir Hatları ferry from Karaköy or Eminönü to Kadıköy on the Asian side takes about 20 minutes and costs almost nothing — less than a metro ticket. During this crossing you pass Topkapı, the Hagia Sophia dome, the minarets of the Blue Mosque, the old city walls, and the mouth of the Bosphorus. No tourist boat can improve on this.
For a longer trip, the public ferry to Üsküdar or the full Bosphorus excursion boat (İstanbul Boğazı Günlük Turları) goes all the way to Anadolu Kavağı at the Black Sea end — a 6-hour round trip passing fortresses, yalıs (waterfront mansions), and fishing villages. The Heybeliada and Büyükada ferries (Princes' Islands) are another option for a half-day escape from the city. All use the Istanbulkart.