The major neighborhoods, streets, squares, and landmarks of Pisa
Pisa is that rare European city where the center of the action isn't th the center of town.
Oh, sure, all the sights you came to see—the world-famous Leaning Tower, the Cathedral, the baptistery, and their related sights—are clustered on a Piazza del Duomo as usual (though that grassy square also goes by Campo dei Miracoli, or "Field of Miracles").
There is, stretching south of this, a classic centro storico, an historic district of arcaded shopping streets and palazzo-lined piazzas peppered here and there with Gothic churches.
Perhaps having the Piazza del Duomo way up at the north end of town dragged the whole concept of the city-center away from the Arno River, because—again, unlike in most cities—there is virtually nothing of interest beyond the Arno's southern banks—just the Pisa Centrale train station, which isn't so much "interesting" as it is a potential part of your travel logistics.
Then again, it might not be; the Pisa San Rossore train station—which receives some inter-city trains as well as those from neighboring Lucca—is in the center's NW corner, a short walk west from the Campo dei Miracoli and all its top sights.
In and around Piazza del Duomo, Pisa's "Field of Miracles" around the Leaning Tower at the city center's N edge