Hi there, lots of good advice above. I will add that the Trevi fountain is much quieter first thing in the morning than later in the day. I would go there first & then walk over to the Pantheon and use RS's audio guide as recommended above. I know tickets are required to enter the Pantheon now, but don't recall how that works, use search bar above? Note that the highly recommended early entry to the Sistine Chapel is very expensive, I seem to recall reading $400 per person, but it's easy to find. Personally, if someone ever drags me back into the Vatican museums, I would be delighted to pay that to be spared the crowds. HA.
I wondered if you've taken the heat into consideration, on cobblestones, tired, need some breaks and maybe a siesta? Consider researching restaurants near the sites you're visiting so you have an idea where to stop for lunch. My last TR has some recommendations for restaurants. I bought a sun umbrella for my next Med trip, you might consider them for your gang.
Day 3 you may have some more time, but whether after the Vatican Museums and St Peters you can handle standing longer or going elsewhere is another matter. (See my comments below about underground museums, if either of them are near your hotel.) I think you will need GELATO! IF you can stand another attraction, Castel S'Angelo is very close to the Vatican, maybe a 15 minute walk. On the highest floor there was a cafe, great views, not sure if it's still open.
You will have more time on Day 4, but whether you have more energy for another museum or standing out in the heat is another matter. We visited 2 subterranean museums in Rome, both of which will be much cooler than being outdoors - one close to the Trevi Fountain, with water rushing under your feet, part of an ancient Roman villa, very cool, (this needs reservations, see in my TR for details, you Whatsapp them a day before). The other just around the corner from Piazza Navona, part of an old Roman theater, just bits and pieces but quite a fun exhibit. They're both explained in TR below. We really liked the tiny pizzeria by the slice mentioned in TR, close to Trevi fountain. https://community.ricksteves.com/travel-forum/trip-reports/rome-was-jammed-over-ny-but-still-fun.
Churches - There are some amazing churches in central Rome, with incredible art work, (you need 1-Euro coins to light up the art) , the interiors tend to stay cool and they could offer a nice break. San Clemente, (needs reservation for the scavi below,) St Ignatius de Loyola & Santa Maria in Trastevere (don't miss the ceiling mosaics!) are all lovely, and there are many others.
Another option I have not done personally, would be to take a golf cart tour around the historic center at nighttime, when all the amazing lights are on the Capitoline museum and down the Foro Romano to the Colosseum. OR if you can stand more walking, take an evening food tour, the sunset tour of Trastevere by Eating Europe gets rave reviews. Have fun!
PS, You asked for other ideas on Day 4, had you considered the Galleria Borghese? It's by far our favorite museum in Rome, but really you may need a break from museums, which is why I suggested the subterranean exhibits above.