The Colosseum is Rome's most popular tourist destination. The structure has been standing for over 2,000 years, and it is a must-see when visiting the city. If you are smart, you will avoid the long ticket lines in front of the Colosseum itself and instead buy your tickets at one of the two nearby Palatine Hill ticket booths. Your ticket price includes Palatine Hill and the Colosseum. Go early in the day to avoid both the heat and the lines. Beware of pickpockets and scam artists outside. They are looking for people who seem lost and confused, and will sell you a phony map or a bogus tour. There are official tours, but they will need to be booked when you buy your ticket. You can expect to spend a few hours at the Colosseum, and don't forget to see Palatine Hill while you're there. Keep an eye out for the hundreds of stray cats that roam around this area. They're usually feral, and they hide all over the grounds.
Although Vatican City, home of the Papal residence, is technically its own country, it is surrounded on all four sides by Rome. It can be fun, however, to mail a letter from the Vatican post office simply for the postmark. You should plan on spending most of the day in Vatican City. Seeing the Vatican Museum, including the Sistene Chapel, does involve an admission fee. This is a long tour, and the lines for admission are also intimidating. The Sistene Chapel comes at the end of your tour of the museum; keep in mind that there is no flash photography allowed in the chapel. Once you've looked around, check and see if the group tour exit is open--it is the door on the right side of the chapel. The one on the left leads all the way back out to the Vatican Museum entrance, while the group exit lets you out right in front of St. Peter's Basilica, which is where you'll want to go next. Admission to St. Peter's Basilica is free, though there is a cost for getting into the dome. Keep in mind that modesty is important; don't wear shorts or low-cut shirts, and if you're wearing a sleeveless shirt, you'll have to bring along a jacket. You can be denied admission if dressed inappropriately.
The Trevi Fountain is another of Rome's major landmarks and is awe-inspiring to see. Nicola Salvi completed the fountain in 1762. It is grand, beautiful and, most importantly, a vantage point to sit down and enjoy the view. You can hear the fountain before you see it, and there are plenty of gelaterias all around. Buy yourself a cool treat before sitting on the steps surrounding the Trevi or even right on the edge of the fountain. There are plenty of people who would gladly hand you a rose and then charge you ten Euros for it, so you should politely decline all solicitations and instead enjoy the breeze and the view.
The Cappuccin Crypt is a piece of Rome you shouldn't miss if your taste runs more toward the macabre. This tiny crypt, located underneath the Church of Santa Maria della Immacolata Concezione, contains the bones of more than 4,000 Cappuccin monks. The monks all died between the years 1528 and 1870, so their bones have been there for a while. Don't expect skeletons, though; these bones have been arranged into artwork, lining the ceilings and walls. In some cases, the candelabras over your head are all made out of bones. Admission is technically free, but the church does ask for a donation at the door.