I have never been to Ascoli Piceno - however, looking at the map, it appears to be about a 4-hour drive from Florence. You would be passing through some lovely places and could take your time stopping in Arezzo, Cortona, Assisi, Perugia, Spello or even Spoleto (I have visited all of these places, and they are very characteristic of the area). You could easily integrate the travel over there into your itinerary and see lots of wonderful places and things along the way. The roads are secondary and many times two lanes in each direction. Though the closer you get to your final destination, the more narrow the roads will become (one lane in each direction.
From Florence, you could also take the train. It is approximately 5 hours via train - the only downer is there are lots of changeovers -