Also note that the days are shorter in October than in the summer and the weather may be chilly. Both of those factors will limit how much outdoor sightseeing you can do, or can enjoy doing.
If you have 9 nights and your top priorities are the Vatican and Pompeii, then plan around those. It is possible to see Pompeii as a day trip from Rome, but your time in Pompeii scavi will be so short that you'll only scratch the surface. If you are real Pompeii fans, travel from Rome in late afternoon, stay 2 nights, spend the intervening day really exploring the Pompeii scavi, then take a fast train to Florence next morning. Once you are in Florence you can branch out to Lucca, Pisa, etc. on day trips -- but Florence itself is worth 2 or 3 days, so you see you'll be running short on time.
Night 1 & 2 & 3: Rome, including Vatican
Night 4 & 5: Pompeii
Night 6 & 7 & 8: Florence, including nearby towns/cities
Night 9: Rome, fly out next day
You probably already know that the train trip from Rome to Pompeii involves changing in Naples.