It's going to be almost impossible to get the flavor of Rome (even with a great private tour guide) in one day. My first stay in Rome fifteen years ago was short and I ended up not liking the city very much as a result of a poor hotel experience and not knowing where to go first.
Since that time I go to Rome about five times a year for one-two weeks, and there is so much to see and do! So I'd urge you to decide what interests you most, e.g., Ancient Rome; Christian Rome; Modern Rome; Museums, Galleries and Music, etc.
Once you have decided, I'd focus on one of those areas and make the most of your day in Rome by getting immersed as much as possible.
Here are some suggestions/favorites for the first two categories (which overlap at times):
Forum, Trajan's Forum, Colosseum and Palatine Hill
The ancient port of Ostia Antica (but that would probably take a half day)
Basilica of San Clemente (which is amazing and the further down you go the further back in time to Mithraic cults)
Basilica of Santa Maria Maggiore
Church of Santa Maria in Trastevere
The part of town, Trastevere, which gives a real flavor of both medieval Rome and modern hip Rome
Vatican including grottoes (save the Museums for another visit)
Great (close) day trips include the gardens at Tivoli and Hadrian's Villa.
Hope this is of some help and that you can at some time spend more time in Rome. It's a glorious city!
If you're looking for a great tour guide check out Italy Driving Guide and ask for Pino. Tell him Lisa B sent you. You will get the most amazing tour of Rome or anywhere else in Italy you wish to go. He's been our Tour Guide each time we visit Italy and we've had a great experience. He's very knowledgeable and the vehicles are very comfy.
e-mail is: firstname.lastname@example.org
The web site is: www.italydrivinguide.com
We are going back in July for about a month and are thinking of trying Central Park so let me know how it was. Thanks.