Not sure how tight your timeframes are for visiting, but I’d budget 20 minutes between both stations just in case. And as a reference for whether or not you may want a taxi, walking to Piazza dei Miracoli from the Pisa station is easily a 20-minute walk.

The trains leave from Livorno Centrale to Pisa Centrale and generally takes about 16 minutes between both points. The trains leave ever 30 minutes. For fare prices, you’re probably looking around 2 EURO one way but may also want to factor cost of going to/from the stations via bus/taxi.

There is a train that goes from Livorno to Pisa. You can check out Trenitalia for for times, prices and travel times as well. As far as getting to the station from the port, you might want to get at taxi.