In keeping with our new "politeness counts" doctrine when talking to first time posters, let me attempt to gently steer you into a rewarding and reasonably priced adventure.
First, there is no such thing as a "small" group of 9. This is a big group to most of the members of this Forum. I can see that a decent amount of time will be devoted to your cruise, but that does leave a lot of unaccounted time. Find others in the group to help you plan things. There is no reason why you should do all the legwork.
You are very late to begin planning for a June 6th departure. Many of the best hotel "deals" and situations have already been snagged by people who have been planning for a year or more. Not a real problem--there are many hotel rooms in Rome and in other parts of Italy--but you will need to get creative. Your first choice (or fifth choice, for that matter) may already be booked for the time you need it.
You NEED a good travel guide. Rick Steve's new Italy 2019 guidebook is a must. Order it from Amazon tonight. You can order it from this website, but it is cheaper on Amazon.com. You might consider ordering several copies--nine people can't use just one book.
https://www.amazon.com/Rick-Steves-Italy-2019-ebook/dp/B07C1NQFQ3/ref=sr_1_3?crid=1BATMMBCWQRGX&keywords=rick+steves+italy+2019+guide+book&qid=1557348943&s=gateway&sprefix=rick+steves+italy+%2Caps%2C296&sr=8-3
Did I say to get someone else to help you?
Find out where people want to go. Get a consensus now so you aren't vilified later when it turns out that you haven't booked something that others may have wanted to see. Find out about physical limitations and food allergies. Gluten free is available in Italy, but pasta without wheat isn't really pasta.
Be realistic about time. Your departure day counts as a travel day. So does the arrival day. So does the flight home day. Using your example above, you are departing on June 6 and going home on July 3. That means you have 3 travel days (6/6, 6/7, and 7/3). This gives you 24 touring/vacation days. Subtract 10 days for the cruise, and this gives you 14 free days to fill. Taking your 2 days before and 2 days after the cruise, that leaves 10 days unaccounted for.
If you can flesh out things a little better, we can give you some suggestions to maximize fun and minimize cost.
You may want to buy a different guide book, as well, perhaps one published by Lonely Planet. Many of the sights/restaurants/hotels will be the same, but many will not. Also, get on Trip Advisor and Booking.com to see what kind of accommodations might be available. Good luck!