In Italy the general rule is the further south you go, the more insane the driving is. My friend in Florence (born and bred Italian, works in downtown Florence and sometimes even drives into work) said that the only place more insane than Naples for driving is Sicily. When he would drive me around Florence, I closed my eyes for fear the next sight would be a Vespa driver on the windshield. It never happened, but it's a well-rehearsed dance for natives.
My DH drove us around Tuscany and to Venice a few years ago, and there were moments that gave us pause, but overall it was a good experience. Neither of us would drive into any city or village, though - it was all Autostrada and country road driving. I would look at a map to see what kind of route you might have from Naples/Rome to Bari and if it's mostly Autostrade/country roads, you might be just fine. I don't know where the Naples airport is located, but if it's on the outskirts you might be fine.
Be aware, driving in Italy can be expensive. Many Autostrade have tolls that are not cheap, not to mention the cost of renting a car (and you MUST take all the property damage insurance available!) and of course gas.
