Wondering which of these two itineraries is the better way to go (they are essentially the reverse of each other)

Itinerary 1

Day 1: land in Rome-train to Naples (or Sorrento? or somewhere that can be easily gotten to Pompeii)
Day 2: day trip from Naples to Pompeii
Day 3: leave from Naples to Anzano (visiting family)(getting there late morning)
Day 4: Anzano-leave for Rome (mid-late afternoon)
Day 5 Rome
Day 6 Rome
Day 7 Rome (possibly Ostia Antica)
Day 8 Rome (with maybe a day trip to Pisa if we're willing to put in the travel time)
Day 9; fly home from Rome

Itinerary 2

Day 1: land in Rome
Day 2: Rome
Day 3: Rome
Day 4: Rome
Day 5: Rome (possibly Pisa) (possibly Ostia Antica)
Day 6: Rome to Anzano
Day 7: Anzano to Naples (Sorrento? somewhere else? )
Day 8: Day trip to Pompeii
Day 9: Naples train to Rome-fly home from Rome (if we fly out at 1pm, is this too tight to do? Do we bump everything up by a day and head from Naples to Rome on Day 8 and spend the night there and head to the airport directly on day 9)

Itinerary 2. In Italy, I would never plan on a flight in a city in which I had not slept the night before. There's simply too great a risk of a train strike or other disruption in transportation for my tastes.

agreed - fly to Rome then stay there for a few days before you try to venture further.
Getting into Rome from the airport is easy. You can take the train, or better yet (my 2c) get a taxi, or have your hotel arrange a private transfer. We did this the last time and it could not have been easier - the private transfer - cost us ~50 euros and was money well spent.

I assume it's Anzano di Puglia. How are you getting there?

If you are willing to move on upon arrival, I'd do so -- why stay in Rome twice if you can do so only once?

Likely driving. Hubby's siblings have done it the past two years and say it's managable.

That's what I thought. If we start in Naples/Sorrento and end in Rome than we're only dealing with two hotels. But we want to do Pompeii from Naples/Sorrento and I read somewhere not to do Pompeii on day one or two of your trip since it's a long day and we'll be jet lagged.

I wish we could fly out of Naples, because that would solve that problem, but there are no direct flights and I don't want to spend 21 hours and 2 stops just to get home (there were others, I just used that one as an example since it's was the longest and worst!!)

Flying Naples to Rome and connecting there for home will take no longer than train from Naples to Rome and then connecting to the airport to get home. There may be a bus or an expensive car service Naples directly to Rome airport. But multimodal last day options puts your flight home in jeopardy from so many unpredictable delays. Spend the last night in Rome or add the Naples-Rome flight to your ticket home. I have done the latter and it was quite easy.

IMO, AJPeabody makes an excellent point. I would also note that Naples is a city of incredible treasures, so you might consider adding a day to Naples. You could, for example, skip Ostia Antica if visiting Pompeii. Extra time in Naples would give you time for Pompeii, the magnificent archeology museum in Naples, and a bit more, whether Herculaneum (if antiquities are your primary interest) or magnificent churches or stunning works of art or pizza....

The pizza in Naples is a work of art.

Naples to me was an incredibly old-world looking city - like all of Italy had in my first visit eons ago - narrow lanes with laundry strung between buildings - kids playing everywhere -unlike most Italian cities lots of residents in the city center IME - On Sundays the many churches attract many and they linger outside after services - kids playing soccer - really animated city. Expect lots of trash around perhaps and not so spiffy areas, etc but Naples is unique IME of being an old-fashioned city.

So there are pros and cons to both itineraries. So for those who have done it, which is worst, doing Pompeii on day two when were likely most jetlagged. Or backtracking to Rome for the last day with an extra hotel for one night.