Chapeau is French/cycling parlance for respect (lit. "hat's off to you sir").
I would want to get out of the city as fast as possible, but if you are based in the centre that isn't too hard if you're looking to ride to the East or South. I find aiming for a target at halfway distance with something of interest is a good strategy. With this in mind, I recommend heading towards either Arezzo or Siena.
In Italy as a general rule you need to be pretty confident. City drivers can be very impatient, although it is the scooters that you need to look out for. Out of town, cyclists get a lot of respect and space IME.