There was no way to go directly from Florence to Bologna in a carriage before modern times. In 1530, a willful traveller who insisted on going straight from Florence to Bologna would travel 2 or 3 days cross-country on horseback or on a mule through the Giogo Pass (note: Italian Wikipedia) (elevation 882 meters).
In pre-modern times, vehicular traffic, such as carts or carriages, between Florence and Bologna would have to go either through Pisa, Genoa, and Piacenza, or else south on Via Cassia to Rome and then back north on Via Flaminia to Rimini and then on Via Emilia to Bologna. But as far I can guess there was extremely limited vehicular traffic between the two cities in pre-modern times. Normal people, even moderately rich people, did not do that.
Once upon a time there was a Roman road linking Florentia and Bononia, modern Florence and Bologna, going through the Futa Pass (elevation 903 meters) and the Raticosa Pass (elevation 968 meters) in the Tuscan-Emilian Apennines. The Roman road branched off Via Cassia at Florentia, and went to Bononia through the valleys of Mugello (on the Tuscan side) and the Santerno (on the Emilian side). The Roman road, like most secondary Roman roads, was intended for infantry travelling on foot; it decayed to nothingness over the centuries, and the route saw little use until the 18th century, because of the steep approach to the Raticosa Pass.
If you can read Italian, the Italian Wikipedia article about the Futa Pass is very much better than the sorry English stub.
Eventually, a new road was built through the Futa pass in 1759; this road would eventually (in 1928) become the National Route 65 (note: Italian Wikipedia). Wikipedia says that opening the new road in the 18th century cut the travel time on horseback or on a mule between Florence and Bologna from 2 or 3 days to one (long) day.
The lowest pass is at 882 meters above mean sea level. On the Tuscan side, Barberino di Mugello (note: Italian Wikipedia, because the English article is useless) is at 270 meter a.m.s.l., so you can expect a rather strenuos climb.
P.S. About that carriage: If you are thinking about something like a stage coach as we see in the movies, forget it. Such things existed in the late 16th century, but they were about as rarely seen as a Lamborghini car in our days, and only filthy rich people could afford them. The normal kind of carriage in the 16th century was basically a wagon with a rounded fabric top and no suspension, or at best suspended on chains or leather straps. Springs were introduced about two hundred years later.
