Rome is probably my favorite city in the entire world to visit.
It's mixture of amazing history, food, culture, and architectural beauty is unique, simply put: there is no place like Rome.
Of course, when a place has as much to see as Rome does, you should do some research on what types of things you find most interesting, so you know where to go.
For any first-time visitor, there are certainly a few must dos:
1) The Vatican: love the Catholic Church or don't, Vatican City is a wonder of the world. While you are there, make sure to hit the usual suspects: St Peters Basilica and the Sistine Chapel really are amazing. But don't neglect lesser known spots, such as the treasury inside St Peters, as the Vatican is full of hidden gems to experience. You should leave yourself at least a half-day.
2) The Ancient Ruins Complex: Rome is filled with ancient ruins, and dozens of spots are worth a visit. But the central attraction is the Foro Romano (Ancient Roman Forum), Palatine Hill, and Colloseo, which are all right next to each other. You can access them all with one ticket, which you can and should buy in advance online to save yourself time. You should leave at least a half-day and savvy visitors allocate a full day to the ancient area of the city and also visit the Baths of Caracalla and Diocletian, as well as the Pantheon, which is a bit of a walk but perhaps the most impressive building in the entire city.
3) The Renaissance Palaces and Squares: The city is dotted with beautiful Medieval and Renaissance palaces, many of which are situated on lovely, famous Piazzas. The Piazza Spagna (the famous "Spanish Steps), Campo Di Fiori, and Piazza Navona are all must visits and easily walkable to each other, as well as the Trevi Fountain, which is no-longer under renovation. The Spanish steps area, particularly, is known for great upscale shopping and that entire area of the city is perfect for just meandering on a sunny afternoon. Piazza Barberini is also quite lovely, and many of the city's best, but still affordable, hotels are found in this area of town.
There are also some hidden gems not to miss that are a bit off the typical tourist path:
1) Circus Maximus: It's right under the Palatine Hill, so it's quite convenient. Most people skip it because its "gone" i.e. there is very little of the former stadium left. But there is something incredible about sitting right in the middle of a deserted stadium-area that could seat 250,000 people, the largest stadium ever created in world history, that shouldn't be missed.
2) Villa Borghese: The villa and its surrounding park are the most beautiful natural area of the city. Close by the Spanish steps.
3) Domus Aurea: It used to be closed off, but now is open to pre-booked tours with guides. This was Nero's palace and it shows; there may not be a more opulent ruin in all of Rome.
4) Trastavere: this neighborhood across the Tiber isn't typically a tourist area; its where real Romans go to eat and relax. Some of the best and most affordable trattorias in the city are found here.
5) Aurelian Walls: for any military history buff, they are a bit of a trek but worth it. Rome built them to defend against barbarians and they only ever fell due to treason, in 410 AD.
6) Temple of Portunus: this tiny little temple complex is right next to the river. What makes it unique is that it was converted to a church early and then ignored rather than used for building materials, so it retains much of its marble and its natural character. See it FIRST because it will really help you understand what the greatest temples would have looked like in their heyday.