You can save money with purchase of advance tickets on the express trains from Naples to Rome and Rome to Florence. However you must be willing to commit to a specific train, date and departure time as those trains have compulsory seat reservations. If you'd rather not do that, you can certainly buy tickets on the day of travel, but you'll be paying Base Fare (full price).
On the trip from Florence to Modena, there's usually one change in Bologna. The first segment is often via Freccia high speed, so the same rules apply as those have compulsory reservations. The second segment is via Regionale so you don't need to buy advance tickets for those as the price will be same regardless of when you buy the tickets.
The trip from Modena to Venice will be the same as the Florence - Modena trip, except the first segment will be Regionale and the second segment will be via Freccia high speed train.
One potential issue with waiting until day of travel to buy tickets for the express trains, is that if you're travelling with a group you may not be able to get seats all together. If you don't mind paying for full fare tickets, you can certainly "keep your options open".