Genoa is well known for its fresh pasta al pesto genovese.
Pesto genovese is typically made of locally grown basil and pinoli nuts which grow on some Mediterranean pine trees and parmiggiano cheese.
Pici pasta is also well known in Toscana.
If you have time and are in Pisa, try to go to Lucca, it’s a small wonderful town. Not to far from Pisa, direction Livorno with children, it might be nice to get to Piombino Porto and get the traghetto to visit Isle of ELBA and Napoleon’s home (Napoleon was deported by the French To Elba). Portoferraio is really nice - fantastic ice creams)
If you want to eat well and cheap in Italy:
Go and buy tomatoes, mozzarella di buffala and prosciutto crudo toscano and some fruits, have some plastic plates and cutlery with you and eat them on the steps of some church.
Try lunchtime menu cheaper than dinner menu
Choose trattorias instead of restaurants
Look that the place you visit are full of Italians and not tourists
Go to village bars where you will have to choose generally between a meat or fish meal for 8 euros or pasta for 6 euros.