it is possible in 3 days
Ciao Mukta,
Florence requires at least an entire day dedicated to it, so does Siena. You could do Pisa in a few hours, and since Lucca is near Pisa you could do the rest of the day there.
That about covers your 3 days.
Renting a car is the best option given your limited time, you don't want to have to depend on bus or train schedules. It should also be relatively simpler this way when traveling with children.
Traveling by train would be the next best option, given that Florence, Pisa and Lucca are well connected. Siena always requires changing trains in Empoli so not as convenient, but still reachable. Buses would eat up too much of your time since most are never direct.
I am not sure where Magnino is, is it in Tuscany?