1st time to Rome

Since you have almost no time to get things booked, assign all four adults a chore. One to research in guide books and on the internet, what to see and how to book tickets or tours. Another to research transportation (are your airline tickets booked?). Another to research hotels, look at Booking.com. That leaves day trips if you have time and restaurants.
There are lots of posts here about Rome. Use the search feature, narrow it down to Forum, then 6 months or 1 year. Also look through Explore Europe in the menu and Travel Trips if you need them.

We are supposed to go end of June to first week of July '24

How many nights are you there? Going anywhere else?

It's all spelled out in the Rick Steves guidebook to Rome - will answer the majority of your questions. For specific sites, you can first search here on the travel forum, and then ask questions for clarification.

Yes, a hotel near the Pantheon would be best. You’ll be walking distance to just about everything and probably won’t need public transportation except on rare occasions.

You should buy tickets in advance for the Vatican Museums, the Borghese Gallery and the Colosseum. Some people buy tickets in advance for the Pantheon but others say it’s easier to just get them there. Google map has the link to the official website for each of these places, you can buy the tickets from the links. St. Peter's Basilica is free to enter, you just have to get there early and get in line.

Some great piazzas include Piazza Navona and Piazza del Popolo.

Other great art museums are the Capitoline Museum, the Palazzo Barberini and the Galleria Doria Pamphilj.

Rome has lots of great churches, some of the best are: Santa Maria Maggiore, San Giovanni in Laterano, Sant'Ignazio, Sant'Andrea della Valle, Chiesa del GesùSaint, Church of St. Louis of the French (3 Caravaggios), Santa Maria in Trastevere and Basilica di Santa Cecilia in Trastevere.

What a nice problem to have! Take a breath and start reading Rick Steves' guide book. Buy the Italy one if you are going elsewhere in the country, otherwise just the Rome book. Make all 4 people read it and as suggested by Horsewoofie, assign each person an area to research and a deadline. You need to be booking lodging ASAP then get everything else in place. There is no real all inclusive pass but advance reservations are essential for the top sites.

We were in Rome in April and were able to get Pantheon tickets by standing in line for about an hour. I think you have to book in advance for the Pantheon on the weekend, though. We booked Galleria Borghese tickets and Vatican Museum tickets directly from those museums in advance, and those tickets can be hard to get, so I suggest booking as soon as possible. It is a large savings from going with a 3rd party vendor. You will probably need to use a 3rd party vendor for the Colosseum tickets (we had been before so did not go this time), but I would go ahead and book. The Trevi Fountain is free and beautiful but be prepared for crowds. Rick Steves' free Heart of Italy walk (available on this website or the RS app) will take you to a variety of central sites including Piazza Navona and the Trevi Fountain. We also took a food tour through Eating Europe (Taste of Trestevere) that was fantastic. There are beautiful churches filled with amazing art throughout Rome that you can stop into as you are exploring. Enjoy!

I would also encourage you to look at less busy suggestions. All the places that you mention will be very crowded, and it will be hot at that time of year, and it might just feel overwhelming. Try to plan time to just walk the streets, and wander into all the lovely churches that nobody else is in. The Borghese gallery is lovely, but you need to book tickets ahead of time.

You might want to take a look at the Rick Steves tours listed on this website. He doesn't run his best of Rome tour at your time of year, but if you look at his itinerary it gives you 6 very full days and you could model your time on his ( taking. in mind you won't have the same transport that he does)

You will probably need to use a 3rd party vendor for the Colosseum tickets (we had been before so did not go this time), but I would go ahead and book.

That doesn't appear to be the case - tickets for the standard Colossuem/Forum ticket are available for virtually every day for the next 30 days.

Do you homework by becoming familiar with the Colosseum ticketing site, then be prepared to buy your tickets once your target date is 30 days away. Note that tickets become available based on business hours in Rome. Note that it's incredibly difficult to score tickets for the underground, attic, etc.