The main train station in Pisa, called Pisa Centrale, has luggage storage so you can take the train from the airport to main train station, leave your luggage there and then do sightseeing in Pisa. The Leaning towe is walking distance but if you want to get there faster you can catch a bus.
Head back there and then on to Lucca. I would suggest staying in Lucca at the point, considering you might already be getting there quite late in the day, that way you have more time to see it.... otherwise, if you don't want to spend the night, I'd suggest skipping it altogether and heading toward Siena from Pisa on the first day.
Siena is a very good base - you can check out some tours offered there and from there to the nearby areas
here. You really do need a car to see the Val d'Orcia and Chianti so the tours are a good way to see those areas.
The places you can do with train or bus from Siena are San Gimignano, Arezzo and Florence....
you can definitely head on to Rome from Siena. Assisi is in a whole different region of Italy, outside of Tuscany. I would NOT ever recommend Assisi as a day trip from Siena, coming back to Siena. If you want to visit Assisi, you should do it as you leave Tuscany on your way to Rome, maybe staying nearby for the night and breaking the trip south to Rome into two parts.
You can check train timetables on
www.trenitalia.com - on Sundays schedules are different but they are still perfectly fine for moving around and seeing the area! You can also plan tours on that day if they are offered. Make sure to spend time seeing Siena as well, so mix up days where you are out of Siena and others where you spend time there.