Hello gina, and welcome to the forum!
Whether you get a private driver or decide to use public transportation the thing that jumps out at me with your plan is that you are choosing to do a lot with your luggage in tow. You arrive in Naples with your luggage and want to tour the museum before heading to Sorrento and then want to tour Pompeii with your luggage and then get a city tour with your luggage before heading to Rome.
Instead I might suggest you get to Naples and then immediately head to Sorrento and gain an extra afternoon's time there. Pompeii is easily done as day trip from Sorrento by train with no luggage and you can choose to get a guide or not. You can then return to Sorrento and relax from a tiring day at Pompeii.
Then return to Naples and do a city tour that includes the museum. You will still have your luggage for this section but I think the museum and city tour are a much more manageable combination then Pompeii and a Naples tour in one day. Pompeii is very large and when I first visited I planned to do Pompeii and Herculaneum in one day, but seeing half of Pompeii took all day and wiped me out. I can't imagine that even a half day at Pompeii and then tackling the chaos that is Naples with the deadline of then still heading to Rome is going to be a great experience. The museum is a shorter and easier visit than Pompeii and the pieces may have more significance to after seeing where they came from.
Also if love pastry check out Antico Forno delle Sfogliatelle Calde Fratelli Attanasio about 5 minutes from the Naples train station at Vico Ferrovia 1/2/3/4. Show up, grab a number immediately and then wait your turn for the best sfogliatelle in Naples hot from the fire oven. Seriously, not to be missed - both pastry and Italian experience - and very convenient to the train station.
My $.02, have a great trip,
=Tod