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.