If there's one thing you have to do when you're in Livorno, it's have an ice cream! At least one. I don't think Italian ice cream - gelato - can be bettered, and there are many ice cream shops, or gelaterias, where you can put my theory to the test.
You might be surprised to find that most gelaterias offer only limited seating, if at all. That's because Italians tend to buy their ice cream and stroll around with it, served either in a cone or a cup. You can do the same, or choose one of the ice cream shops that have a few tables and chairs on offer.
Ice cream is very popular in the summer, of course, but it is made and eaten all year round. The cheapest cone or cup size allows you a choice of two flavours, a hard decision when faced with a couple of dozen. The more you pay, the more flavours you can try on one cone! Add whipped cream (panna montata) for a real treat.
If you're a serious gelato fan, you could plan a tour around Livorno's gelaterias, trying out different flavours, and combinations of flavours as you go. Let me know about your experiences!