We flew into Pisa and had a very short layover before we took a train to the Cinque Terre. We hopped into a taxi at the airport, had them drop us off at the tower, spent a few wonderful minutes there, then grabbed another taxi to the train station.
If I recall correctly, the train station was not too far away from the tower. Most train stations have luggage checks.
I would arrive at the train station and leave my bags there if possible, then return to the train station to pick up my bags and take a taxi from there to the airport. The airport into town was a longer trip than going from the train station to the tower, so you'll save time and money by leaving your bags there than going out to the airport twice.
Rick's book had pretty good instructions on all of this, so consult there for more details. My memory is not as good as his information!