You (like all of us) have limited time and I assume you don't like to stand in lines, especially in the heat of Rome. Ditto the suggestion right above mine about pre booking your Vatican visit.Do it now. nothing in your imagination could accurately describe the length of the lines so do get reserved tickets.
There is also terrific website called www.roninrome.com. click on "Sites and Attractions" and you will see very helpful and insightful suggestions regarding visiting the Vatican, including suggestions on what days and what hours to visit.
We visited in April in the middle of the week at a supposedly slow time, and although we reserved tickets and so had no line to enter, parts of the Vatican Museum were literally wall to wall people. I advise you to do a little bit of research, Google "famous art at the Vatican museum" or "Best of the Vatican Museum". take a look at what interests you so you can plan your visit and hit the highlights. I assigned each member of my family two of my favorites(ha ha), and each one of us found a photo of the item on Google, and put together a few pieces of information on each piece of art, easily found on Wikipedia, and we all share the information so we knew what we were going to be looking at.Is this the advice of art historian would give you? No, but it was a great way for us to see 8 different pieces of wonderful art, and avoid the crowds of tourists.
The Ron in Rome website also has other helpful information. If you are traveling with 3 other people in your family, maybe they can help you doing the research by checking out that website! Lastly, there are a million other sites to see in Rome, but if I could put a quick plug in for the Pantheon, along with Rick Steves free audio guide, I would advise a visit, which can last only 15 minutes if you are pressed for time but it is really unforgettable. Enjoy!