Rome is a city full of monuments and interesting things anywhere, so it is worth spending a lot of time wondering around. Anyway, if you have no time (or money) to spend, here is a short guide to be able to see all the most famous and interesting parts in only three days. It is expected that you have a map to lead yourself around and a guide explaining the things at which you are looking.
Day 1, The ancient part. Take a look at the oldest part of Rome. Arrive in Piazza Venezia (it's quite easy since a lot of buses go there) and go walking to Piazza del Campidoglio. From there you can find narrow stairs going downwards to the Roman Forum. From the path passing through the Roman Forum, there is another path on the right going upwards to the Palatine Hill; go there. After a while you arrive at a place where you can buy the ticket to get in. With that ticket you can go to the Colosseum as well and the price is the same as the one for the Colosseum only, so it is worth buying. Also, try to avoid the huge line to buy the tickets at the entrance of the Colosseum. Get in, take a look at the Palatine Hill and, after you have finished, come back to the path in the Roman Forum. Go on until you arrive to the Colosseum. Now you can get in with the ticket you purchased before. Come back to Piazza Venezia through via dei Fori Imperiali. Here you can get the bus to come back home, which should be enough for one day.
Night 1, Trastevere. Go to Trastevere at night. It's a nice area and it is better at night since it's a common place for nightlife in Rome. It is better to go there with the tram 8; you may walk from Piazza Venezia as well, but it is going to be quite a long walk! There you can find lots of good restaurants, some of them cooking traditional Roman meals. Moreover it is full of pubs where you can find good Italian or imported foreign beers, mainly from Germany.
Day 2, The more modern part. Go to Piazza Venezia again but this time go towards via del Corso. Keep on walking straight as long as you don't arrive to Piazza del Popolo. You may climb up the stairs to go on the hill next to that, where you can enjoy a nice view of Rome. Come back to the square and go towards Piazza di Spagna. Once you are there find your way to Trevi fountain and the Pantheon. After this you have walked enough for today.
Night 2, Campo de' fiori. Go to Campo de' fiori square. You can get there with the tram 8. It is a typical place to meet each other in Rome and it is always full of people at night. Here there are a lot of pubs all around: you can get a drink and enjoy it sitting somewhere in the square. From there you can go to Piazza Farnese to take a look and then to Piazza Navona, where there are several pubs.
Day 3, Vatican. Go to the Vatican church by subway. Go inside,and if you are brave you may want to climb up the stairs to the top of the cupola. After it you can go to the Castel Sant'angelo. It is worth getting inside. Events are quite common there, so you may find an exposition of paintings, jewels or whatever else. If you want to go to the Vatican museums as well, you'd better wake up early in the morning and start lining up as early as possible. Once you are in, there are so many things to see that it would be worth spending the whole day inside. If you want to also see the church on the same day you'd better speed up!
Night 3, Testaccio. Recommended only for young people that want to enjoy themselves going to a club. The place with the highest concentration of clubs is Testaccio. Since there are only clubs, you'd better go there on Friday or Saturday night, otherwise you might find just a peaceful place with nobody around. There are many kinds of clubs: dance, hip-hop, rock, etc.
Obviously there are lots of other things to see in Rome, but for just three days this is definitely enough!
This is a usable itinerary. It explains how to get there and touches on all the major points along the way. An adventurous person could use this article, but please plunge forward and help it grow!