Correct. Just buy your tickets at the station, validate it in the little yellow box and get on. Except for Eurostar, EuroCity and maybe ICplus trains, the tickets you buy are "open". The ticket will say La Spezia Centrale to Pisa Centrale but there is no train number or seat reservation on the ticket. It's good on any train making that run. That's why they have the little yellow box. Most 2nd class tickets are open which gives you a lot of flexibility of when you want to travel. Here's what the schedule looks like.