The city has extensive bus lines, metrobus lines, 2 underground metros, two tramvays, a network of ferry boats, and tiny mini-busses called Dolmus (Dol mush). Of these, the only ones that most tourists need to know about are the Ferries and the European Tramvay.
The ferries are great and can be picked up in Eminonu, Karakoy, Kabitas, Besiktas, and all the way up the coast. From the european side, they can drop you in Kadikoy, Bostanci, or Uskudar, with Kadikoy being the most common. They are really evocative and romantic, (although FREEZING in the winter) and they have unreasonably good tea on the top decks. It would be a bit insane not to take at least one ferry ride in Istanbul.
The tramvay runs from Bagicilar to Kabitas (with a funicular up to Taksim) and along the way hits all the major tourist zones: Sultanahmet, Gulehane, Eminonu, Karakoy, and Tophane (where you can find the art museums). It's usually pretty crowded, but runs often, quickly, and cheaply (2tl for a ride). Please note that during crowded times, there have been incidents of groping on the Tramvay- if this happens to you, do not be afraid to shout "Yapma!" which means "don't" in Turkish, or to make a scene. Shame is a real force in turkish society and will cause the harasser to beat a hasty retreat.