Pisa's weather is influenced by its proximity to the sea. In late fall the precipitation starts, as you can see from the charts below.
Spring is a fine time to visit Pisa. The Regatta della Antiche Repubbliche Marinare, or regatta of the old Maritime Republics, is held in Late may or early June, usually in fine weather. Eight man crews race against each other along the River Arno; don't miss the medieval parade along the banks of the river. (See Pictures from the 54th Regatta)
The rest of Pisa's big festivals are held in June, the Festa di San Ranieri, when the Arno is adorned with torches, occurs on June 16th, and the next day there is the Regatta di San Ranieri.
Sure, you're going to get off the train at Pisa Centrale and run to the Leaning Tower as if that were the only attraction in the city. You'd be wrong, but we'll aid and abet your efforts with a video: Getting from Pisa Train Station to the Leaning Tower in a Hurry.
Pisa and nearby Florence both have airports. To get between the cities and the airports, see How to Travel from Pisa to Florence by At Home in Tuscany, a blog from Pisa.
Hostels and Inexpensive Lodging in Pisa
Como | Genoa & Cinque Terre | Florence | Rome | Naples & Amalfi Coast | Pisa | Venice
Sardinia: Alghero | Oristano | Cagliari
Sicily: Palermo