We've only been to Rome three times, so some of the other posters likely have much more experience from which to draw. You've been given some great advice, to which I will just add the following:
1) If there is a site that is a 'must-see' in your mind, consider doing it early in your visit, so you have more time over the following days, should you run into a problem seeing that site.
St. Peter's comes to mind for me............it was on our THIRD trip to Rome that we finally got inside St. Peter's. First time it was closed for a private mass, and you can only imagine how long the lines were as more and more folks lined up waiting until it reopened for the public (picture busloads of nuns, busloads of tourists, etc.) The second time, the beatification was announced (after we had already made our trip reservations, etc.). While it was truly unique seeing all the to-dos for that event, no luck getting into St. Peter's due to all the related events. Finally, our third visit, I told my husband that if we didn't make it in, I might have to once again becoming an active Catholic.....maybe a message was being sent to me...LOL! But, I had my spouse take a photo of me doing the ole 'thumbs up' as we walked thru the door of St. Peter's on the third visit to Rome.
If you have Rick's book for Italy (mine is a few years old), there should be contact in for a guide named Francisca (sp?)....whatever she charges (within reason) definitely Hire Her for maybe two hours to take you thru the ancient sites...Rick uses her for his group tours, and she is absolutely incredible!!!! She makes the site come alive, her English is perfect, and she is a fun person, too (Likely others will chime in about how wonderful she is.) I assume she is still guiding...it's been a few years since we were in Rome.
2) For the Borghese, if you happen to arrive early, just politely check with the ticket office to see if perhaps you could enter a bit early. They allowed us to do that on one of our arrival days, when we booked tickets later to allow for late plane arrival, etc. I kindly explained we didn't want to miss it due to a late plane, but that we were also beginning to get very tired (jet lag) and the gentleman made a switch to our tickets and let us join the group that was leaving in 5 minutes.......despite signs posted that they were sold out and tickets were not available for something like three days later. We appreciate his kindness.
3) As another poster said, use the public transportation. The subway (or whatever they call it) is not extensive....Francisca explained that every time underground digging takes place in Rome, more great relics and discoveries of ancient civilization are found, so the subway will not be expanded in a hurry. But, taking that to the Vatican area (and then it's still several blocks of a walk, but very do-able) will save you gobs and gobs of leg time. In Rome, there is so much walking, you will want to save the walking for when you 'have to' walk. We walked all the back to our hotel (in general area where you will be staying) from the Vatican and I don't think we would choose to do that again...should have taken the Metro subway system.
4) ........and as a previous poster said, definitely have a plan, but also go with the flow....we've always had to make little alterations to our plans in Rome........and I am a 'planner' by nature. I usually nail down an itinerary by morning, mid-day, afternoon, and evening, and usually 1/4 to 1/3 of that gets wiggled around or changed once on the ground.
Have fun!! I hope you will also have time to get into the countryside, too.
