The closest airports to Prato are Pisa airport (where lowcost airlines like Ryanair and Easyjet fly) and Florence airport (very close to Prato but more expensive). Bologna airport can be a good option too, with many airline companies flying.
From Pisa Airport:
Train station is just outside the airport. In order to arrive to Prato you need to change 1 or 2 times. It costs 8.60 euro. It takes altogether less than 2 hours. For timetable, look at here: http://www.ferroviedellostato.it/homepage_en.html