Shanghai's quick and efficient subway system—called the Shanghai Metro—is an excellent way to get around town, and the network is growing exponentially every year. English maps and exit signs abound, and the single-ticket machines have an English option, too. In-car announcements for each station are given in both Chinese and English. Keep your ticket handy: you'll need to use it at a second turnstile as you exit at your destination.
Refillable transport cards costing Y20 are available from the info booths at metro stations. Add as much money as you like, and use them to pay for taxi, metro, ferry, and bus rides. They can also be added to Apple Pay on later model iPhones, which can then be used in lieu of the card itself. The transport cards aren't discounted, but they'll save you time you would have spent joining queues and fumbling for cash. To purchase, say, Wo yao yi zhang jiaotong ka (我要一张交通卡) which means "I want a transportation card."). Shanghai is a sprawling city with large districts, but the downtown area is fairly compact, and the subway reaches nearly all places you'll want to visit.