Hello lira and welcome to cruise-chat;
first let me tell you how envious I am; I've wanted to cruise on Prinsendam since I first heard about it--it's what I consider the ideal size. And your itinerary looks facinating. Of your ports-of-call Livorno and Monte Carlo are the only two I can offer advice on.
In Monte Carlo we did a walking tour in the morning before the heat of the day, and then a motorcoach tour along the coast in the afternoon (all ship-sponsored as it was our first time there).
From Livorno the usual choices are Florence and Pisa. Don't try to do both; you won't see much of either. My advice: spend the day in Florence and save Pisa for another trip.
Unless you are familiar with a city I would stick with the ship-sponsored tours--don't set off on your own, unless it is to rent a private car and driver, which can be very expensive.
Book in advance. Half the fun of a trip is the planning. Look over the offerings on HAL's website or send for their brochure. Pick things that look interesting to you.
The biggest mistake my wife and I made on our first European cruise was trying to do too much. Our thinking at the time was, "who knows when we'll get back here--let's do as much as we can". We saw a lot but came home exhausted. If we had it to do over again we would choose a more leisurely pace. I would suggest either a morning or afternoon tour; not both. Morning is usually best; it is cooler and you can go back and relax on the ship. Save the all day tours for something special, like Florence.
I could go on and on but I'm sure others have much more to offer you.
Have a wonderful trip. You are on a great ship and a great cruise line with a facinating itinerary.
Keep us posted.