I have planned for us to take public transportation from Florence to Pisa (just for the Field of Miracles) then continue on our way to our destination of Vernazza. What I didn't take into account is we will be traveling with our luggage (family of 4 -- two teenagers). Does this limit us to only taking a train (RS had mentioned a baggage check at the main train station in Pisa)? Has anyone traveled by bus (Florence to Pisa) and is there a baggage check at the bus station? Then would I continue by train to the Cinque Terre? Alternately, can a train ticket be bought for the full trip (Florence to Vernazza with a stop in Pisa to get off and back on)? I'm just wondering if anyone has done something like this and, if so, how did it work for you (or what would you do differently)? Thank you!
I'm not sure why anyone would travel by bus from Florence to Pisa? The train is direct, frequent, and inexpensive. If you have bags, you can leave them at Pisa Centrale station - I guess if the bus drops you there too you could also leave them there. (No requirement to have taken the train to use the bag drop.)
You'd need to buy separate tickets for each train segment. But you can buy them ahead of time. If you are buying regional tickets (not reserved), you don't need to buy them until you are sure of your dates. They cost the same whether you buy them three months ahead or 20 minutes before you get on. (But paper tickets bought from a machine or an agent must be validated at the station before boarding!). However, trains with reserved seats tend to have cheaper tickets available early and be expensive last minute, so it helps to buy them earlier if you are sure of your plans that day.
You can also buy train tickets on your phones.
I have taken a bus from Firenze S.M. Novella to Pisa but only because a strike halted trains to Centrale (where we needed to switch trains to Monterosso). The bus we took was also a shuttle to the Pisa airport and not to the main train station; we had to take local transport to Centrale but that wasn't a huge deal. I've no experience with buses from Florence to other locations in Pisa.
For what you want to do I'd definitely take a train to Centrale, check your bags, head off to Piazza Dei Miracoli, and then back to the station to pick up a train to Vernazza. As posted above, you'll need to have separate tickets for the Florence> Pisa leg, and Pisa> Vernazza leg.
You have mixed up two different things.
When Rick (or anyone else) says there is a "baggage check" at a train station, he means that you can leave bags at the station for a day or less, for a fee. This is sometimes called "left luggage." But on buses and trains between cities in Europe, there is no "checked luggage" like there is on an airplane (or on some of the long distance trains in the US).
On a train, you carry your luggage aboard and store it above your seat, between the seats, or in racks at the end of the cars. Details and photos here: https://www.seat61.com/luggage-on-european-trains.htm
On a long distance coach (say, from Rome to Sorrento), there is luggage storage under the bus; with the help of the driver, you put your own bags under the bus just before boarding. I don't know if the bus from Florence to Pisa has luggage storage underneath, or is more like a city bus, which would mean carrying your bags on (this is what you're concerned about, and I do understand, as these don't have luggage space).
The shuttle bus we took from Florence to Pisa airport had luggage storage underneath but there are probably regional buses as well that I can't answer for.