Michael, you've done a great job so far booking flights and lodging. I disagree that a travel agent can give you better advice than you and your family can find on your own. Travel agents have not necessarily seen what they are selling. I had a very bad experience with a travel agent, so am not a fan of travel agents, especially when I can plan and book sights/experiences myself.
First of all, get guidebooks. Rick Steves guidebooks are good, as are others. RS guidebooks are on sale now on this website. There are all kinds of websites and blogs for ideas, including Explore Europe/Italy in the RS menu. There are many suggestions from previous forum posts and trip reports; use the search feature at the top of the page.
If you want guided tours, in Rome and the Vatican I enjoyed https://www.takewalks.com/rome-tours/ They don't have tours for Sorrento but you can find local guides for Pompeii. Food tours are fun; https://www.eatingeurope.com/rome/ is just one example. There are also free walking tours for an outdoor overview of an area.
If you can, assign each day to a family member. Keep in mind that all six of you may not want to see the same things; so break up into smaller groups during the day and regroup over dinner to share your adventures. Or if family members are as indecisive as my sister-in-law was, I'd give her a very narrow choice of what to see each day, then I'd book what she decided. (More work for me, but we saw a lot that we both wanted to see. I love the planning part.) Look at Google maps to plan where to wander to. Just don't overschedule each day so you have time to enjoy the experiences.
Keep asking questions. When you plan your itinerary, allow time for transit, food and breaks.