Florence to Barletta

You could split this up. The best itinerary looks like a Frecciarosso to Bologna connecting to a Frecciabianca to Barletta. Since the train to Bologna is only 35 minutes, you could go Standard class for that leg, then travel 1st class on the longer 5 hour train to Barletta. With Super Economy tickets, that only costs 10 EUR more.

ViaMichelin says its at least a 6 1/2 hour drive. I don't see how you can do it any faster unless you have a private jet.

Ryanair flies nonstop to Bari from both Pisa and Bologna, and you could take a train from Bari to Barletta. When you factor in the time to get to the airport, check-in, transfer to train station, etc., I'm not sure it would be much faster. It also involves more changes and may be more expensive. It is another option though.

I haven't done this train ride the whole way, but I did go from Caserta to Bari and it was a comfortable train ride. Second class on high-speed trains is nice enough that there's no need to buy first class (or any higher class of service) unless you can get a good advance purchase fare (i.e. one that's comparable to a second class advance purchase fare).

Thanks to everyone for your suggestions