Train travel is the way to go. We went to Florence, Venice and Milan (for getting to the airport) and took the "slow" train each time. Don't bother with the awkward online interface. Go to the train station and use the ticket kiosk. Click on the Union Jack (if you speak English) and then search for your destination. What you'll find first are all the bullet train fares which are Very Expensive! but click on "search other options" and you'll see the 1st and 2nd class fares and the slower, regional trains. Going from Bologna to Florence takes about an hour and a half (on the regional train). Why would you need to get there any faster? Plus, the price of a regional train ticket is not that low when you factor in a family of four. The stops are always announced and the trains are very clean and comfortable, not at all like our Metro North or Amtrak trains in the US.
The electronic board at the station tells you the track numbers. Before boarding, validate your ticket at the machine located at each of the gates. But don't freak out if a conductor never comes by and takes your ticket. We could have saved even more money if we had just skipped the ticket purchase altogether, if we had only known when we'd get asked for the ticket and when we wouldn't. But you didn't hear this advice from me!