Rome and south of Rome

If you want to explore the Amalfi coast, Sorrento, and maybe the island of Capri and Pompeii as well, you can't do all of that from Rome. You'd need to go several days back and forth, and a lot of time will be wasted getting there and back, and you will spend money on unnecessary train fare as well. Sorrento is probably the best base for your intended itinerary.

My suggestion is going to Sorrento immediately after landing since your flight is likely getting to Rome in the morning. That will add a couple of hours to your journey, but that way you spare yourself an extra hotel change. Spend at least 3 nights in Sorrento (but 4 would be better). Then come back to Rome for your last 3 nights before flying home. Not sure if you have the time to add a day trip to Florence, if you are going for a week only. Two locations are enough. Rome alone deserves 3 nights (2.5 days) at the very least.

Ditto to what Roberto said: Rome is too far away to use as a base for the Amalfi/Pompeii. I'll second his suggestion for basing in Sorrento.

2 locations (Rome and Sorrento) are about the max you should do with just a week to work with; I wouldn't try to cram any day trips from Rome into the short amount of time that you have. There is a LOT to see in that city so save Florence and other locations north for the next trip.

Saturday. Arrive in Rome. Train via Naples to Sorrento.
Sunday. Day trip to Capri. Or Positano.
Monday. Day trip to Pompeii.
Tuesday. Train to Rome. Stop in Naples on the way.
Wednesday. Rome sights.
Thursday. Rome sights.
Friday. Rome sights.
Saturday. Depart from Rome.

3nts vs 4 nts depends on where you prefer to focus.

Rome day trips, Ostia Antica, Tivoli, Hadrian's Villa

David'd suggestion is good, except that he doesn't give you much time for the Amalfi Coast (like Positano), which you said is high in your agenda, and actually proposes to choose between that and Capri.

I think you have time for both, if you limit your Rome visit to 3 nights only. Also, the day you visit Pompeii, you can also visit Naples in the afternoon/dinner. So you'd basically visit Pompeii in the morning, then in the afternoon head to Naples for a quick afternoon visit and a pizza in the evening. Or you could do Naples in the morning, have a pizza lunch, then come back to Pompeii for the afternoon. Then in the evening you return to Sorrento.
The last train back to Sorrento departs from Naples at 9:41pm. If you do Naples first and Pompeii in the afternoon, the excavation site closes at 7:30 pm in summer (but last entry is at 6:30pm I think). 3 or 4 hours are enough for Pompeii.

Thank you all for your ideas. It helps me a lot in my trip planning.

Can anyone of you suggest what's the most convenient area to stay in Sorrento? Also if I do all of these, can I do it myself using the local transportation? or join the shared tours for Rick Steves with Mondo Guide. Can you name a place where I can get pizza in Naples?

You are very late to be booking in Sorrento. Prepare at least seven choices during your research.

Having been to St. Peter's recently, I want to mention how focused they are on pilgrims this year. If you know any group that might be marching through the special street corridor to sing and pray near the architectural "crossing" in the Basilica, you might get a special thrill unrelated to your Papal audience. (Don't ask me for information, our visit wasn't based on "faith".)

You are better off staying in Naples to see Pompeii, but you need to allow time for the Assisi visit. Which is more important to you? There are plenty of important churches in Tuscany, for example Orvieto, also easy by train. Siena would be much more time, but they might have their famous Duomo floor uncovered during part of October. (Look it up.) You have too many ideas for such a short stay. Start cutting things out now, or you will spent your entire visit on the train.

As Roberto says you may be happy with only 3 nights in Rome, and in fact, that would be my preference also. (Hence my subnote of 3nts vs 4nts)