I am a big fan of starting a trip to Italy in Venice (getting over jetlag there and not having to deal with road traffic and being able to easily walk everywhere or take vaporetto rides). It's an hour and a half by train from Bologna, so a bit quicker than the 2.5 hours Bologna to Rome and immediately being plunged into one of the busiest cities in Italy.
The train travel times I'm giving here are the fastest, not what are the most convenient. Get the Trenitalia app and see what works for you. Sometimes I start looking on Google Maps for the trains, because it's easier, then switch to that app to make sure.
Going straight to Venice (or to anywhere except Bologna itself) does make for one long first travel day, but it's doable and we have done similar even longer first days and found it well worth it. I would be inclined to steal a night from your two final nights in Bologna and add it to Venice.
Then I'd take the 4-hour direct train from Venice to Rome, then the 1.5 hour train to Florence, then the half-hour train back to Bologna.
I don't know what time your flight to Italy is or when the kids' last day of school is, but see if you can get your family a couple hours onto Italy time before your leave home. The last time we took our son, daughter-in-law, and two grandchildren to Italy they did this and were glad they did. Remember to have meals at the right times, not just sleep.
Search the Italy forum for more details and post more questions as you have them!