Yes many places are free first Sunday of the month, but also very busy on those days. Essential to buy tickets for the Uffizi and Colosseum online ahead of time in order to skip the queues. The queue for the Uffizi in particular is very long without a pre-booked ticket.
See this page for Uffizi booking details:https://www.uffizi.it/en/pages/how-to-book-tickets
This is the official site for booking Colosseum tickets:https://www.coopculture.it/en/colosseo-e-shop.cfm
For the Vatican I can recommend the ‘Pristine Sistine’ tour by Walks of Italy. It is expensive - but in my opinion totally worth the money. Superbly knowledgable guides, early admission to the Sistine Chapel, and straight into the Basilica after the Vatican museum tour, without having to join the often extremely long queue. My only worry is it may be fully booked for your dates by now :( If that’s the case, you should book Vatican tickets in advance as you will be in a long queue otherwise.https://biglietteriamusei.vatican.va/musei/tickets/do?weblang=en&do
No need to book for the Pantheon.
In Florence be sure to see the Duomo and the Baptistry. Also walk up to Piazzale Michelangelo for stellar views of Florence - sunset is a great time to do it. If you’re interested in renaissance art check out the Brancacci Chapel, it’s a true hidden gem.
In Rome you should try to see the Borghese Gallery. In my opinion the best gallery in Rome (and there are a lot of galleries there!) Pre-booking essential, and unfortunately by now the tickets for your dates will be long sold out. However, there are almost always no shows, especially on first Sunday of the month, (when entrance is free but pre-booking still required) so you should turn up there and try for a couple of those. I suggest arriving there at around 4pm for the 5pm entrance slot. Get a coffee, and start queuing no later than 4.30pm. Keep fingers crossed!