For anyone interested in western history or art, there is no destination better than Rome! And while there are different modes of transportation available (driving in the city is not recommended), the best way to see the Eternal City is on foot. There are many accommodations centrally located in the city, and it would be advisable to stay there. From your hotel, most of the city and what it has to offer will be a short walk away.
Three full days in Rome is best, anything less would not be enough. And you can certainly do more without being bored, too! But if you only have 3 days, here are some recommendations, and keep in mind, these are full days - start early!
Day 1: Ancient Rome. To give you a taste of what's to come on Day 2, you may want to start with a visit to St. Peter in Chains (San Pietro in Vincoli), a 1500 year old church that houses not only the chains that bound St. Peter, but also Michelangelo's Moses. A fantastic piece of sculpture. Then continue a few blocks away to the Colosseum. This majestic ruin is well worth the tour inside. It will probably be early afternoon by the time you leave the Colosseum, and that should provide you with enough time to explore the Palatine hill (your Colosseum ticket allows you entrance to the palatial ruins on this hill). And don't forget the Roman Forum! It closes at sunset, so give yourself enough time. And if you don't have a book that describes the buildings in the forum already, pick one up from a vendor outside. After leaving the forum, climb up the Capitoline hill and watch what the setting sun does to the forum & colosseum.
Day 2: The Vatican. Give yourself a full day for the Vatican. Go to the Vatican museums first. The line grows very long very quickly, and there is a lot to see inside. It is impossible to use hyperbole to describe the Vatican museums. They have relics from all parts of the world and from all eras of time. Make sure you buy a guidebook before heading in, as not many names are given, let alone any history or insight, and you'll want to know what you're beholding. The museum leads you eventually to the Sistine Chapel, whetting your appettite first with the Raphael rooms. Spend time admiring these great works from the talented Renaissance painters. After the museums, head over to the largest church in the world, St. Peters.
Day 3: the Rest of Rome. There are plenty more sights to behold in Rome, and most of them within walking distance. Check out the Trevi fountain, the Pantheon, Piazza Navona, then over to Castel Sant Angelo or to Piazza Del Popolo and The Spanish Steps.
Be sure to take time to enjoy your surroundings, too. The atmosphere, the culture, the food, the architecture, the people. It's an amazing city, and it deserves your time and respect. And there's no better way to see it than on foot.