The easiest way to get around town is by using the subway, which is extensive, and good value, although like everything else in this city, it does get ludicrously busy during the rush hour. The buses are unpleasant, but the taxis are very good, and many of the drivers will accept credit cards for payment.
It's not necessarily a good idea to hire a car if all you plan to do is stay in the city, but if you are planning to tour the surrounding areas, it is a useful thing to have.
The Huangpu river runs through the centre of the city, and there are various cruises and ferries along and across it that give you excellent views of the skyline, and of the Bund. The best time to take these is at night, when the buildings are lit up in a dazzling display of lights.