Taxis and hired cars are the practical option for getting between the east and west banks and reaching sites on the outskirts, such as Dair Mar Elia or the Nineveh ruins. Agree on prices in advance — meters are not in use. The city is spread across both banks of the Tigris and connected by several bridges; the crossing you use will depend on which area you are visiting.
The Old City on the west bank is compact enough to explore on foot once you are in it. The lanes are narrow, often only wide enough for pedestrians, and a walking exploration of several hours will take you past most of the main historic monuments. The east bank is more modern and spread out; for Nineveh and the Mosul Museum, a taxi or arranged vehicle is the sensible choice.