Although there is much to see and do outside of central Rome (Appia Antica and Ostia Antica are two examples of many), there are so many things to do in central Rome that a whole itinerary could consist of a single neighborhood. For this reason, we'll focus primarily on the three most historic parts of Rome: the old city center, the Colosseo district, and the modern city center.
The old city center, located between the Tiber River to the west and Piazza Venezia to the east, is an area replete with grandiose buildings and piazzas of various eras, with representations of both ancient Rome and the Catholic structures of the last two milleniums. The most famous structure is the Pantheon, the only Greco-Roman creation still intact and in use today (built in 27 BC), and known for both its spectacular dome and the hole in the ceiling that actually allows rainfall in. For those who desire a taste of open-air antiquity, the Largo di Torre Argentina has the remains of four Roman temples and Pompey's Theatre. Perhaps what makes the old city most alluring though are its many piazzas, some small and hidden with dedicated monuments, while others serve as tourist havens. The two most popular squares are Piazza Navona and Campo de Fiori, both of which serve as socio-cultural centerpoints of the district. Piazza Navona retains the original shape of Stadium of Domitian, where the barbaric Roman games took place. This huge pedestrian piazza is bordered by regal palaces and churches (Palazzo Pamphilj and Sant'Agnese in Agone are two of the most lovely), while its three fountains are quite the spectacle. Campo de Fiori is another square nearby that is notable for its market and the hooded statue of Giordano Bruno, a martyr of modern science. Attractive, lesser known squares include Piazza di Pietra (known for the surviving facade of the Temple of Hadrian), the Piazza di Sant'Ignazio (home to the Baroque church of the same name), and the Piazza della Minerva (the Gothic Santa Maria sopra Minerva), among others. To be honest, the old city center has a nearly endless succession of piazzas and buildings of interest, making it a perfect place to walk, walk, and walk some more as you court a sense of child-like wonder.
If you really want to see what remains of oft-romanticized Rome, then head east of the old city to the Colosseo district. Two massive structures that are etched forever in history are the Colloseum and the Roman Forum. The Colloseum is the largest amphitheatre ever built and is worth the entrance fee to view this treasure of the Roman world; take some time to also see the Arch of Constantine nearby. The Roman Forum is what remains of the imperial Roman seat of affairs; it's many temples, basilicas, and more bely its religious, political, and artistic significance, while the ruins of villas on adjacent Palatinate Hill can also be seen with the same ticket. If you go to see anything, see the Roman Forum; there is no more salient space in regards to Roman history. If you go to see anything else, try Capitoline Hill. There are four museums situated around the Piazza del Campidoglio at the top of the hill, all of which specialize in Roman art and artifacts, complementing an excellent view of the city. Beautiful churches like San Clemente and San Stefano Rotondo also can be found in this district.
Another district of particular note is the modern city center, which begins at the Termini bus station to the west and concludes with the Nomentano district to the east. Most tourists will spend their time here traversing the roadside cafes along Via Veneto or ogling the Trevi Fountain; despite being pretty consistently packed, the Trevi is a must for its sculpture of Neptune and the generally grandiose scale. The Palazzo del Quirinale is a few blocks east and is almost as quickly reached on foot as in a vehicle. Serving as the current home of the Italian president, this regal building sits on a wide square and also has interior gardens that garner a lot of attention from tourists (not worth it in our most humble of opinions). There are also a few piazzas of note in this more contemporary part of the city; Piazza Barberini and Piazza della Repubblica. Barberini is the starting point of the popular Via Veneto and is home to the fantastic Palazza Barberini, notable as the official National Gallery of Ancient Art. The Piazza della Repubblica also gives you a chance to view antiquity at both the Palazzo Massimo and Rome's National Museum at the Baths of Diocletian (more ancient Roman art and sculpture), while the Santa Maria degli Angeli is a church right next door. You'll also find the Termini bus station here.
Other neighborhoods are not without value. On the contrary, some of the most authentic travel experiences in Rome can be had just over the Tiber River in Trastevere, where medieval houses encroach the cobblestone paths and the majority of expats and international students spend their time. There are a number of beautiful churches here, including Santa Maria, Santa Cecilia, and San Francesco a Ripa. The cafe and bar culture here is also top notch. Just north of the old and modern cities are Italy's two most famous squares, the Piazza del Popolo and the Piazza di Spagna. The Piazza del Popolo is Rome's largest square and is the most popular meeting spot for large groups (especially on occasions like New Years) in the city. Ignore the vendors trying to sell you flowers and enjoy the many monuments and churches on on the square, which is quite close to the Spanish Steps and Piazza di Spagna. The Spanish Steps are the largest staircase in Europe (usually strewn with tourists), terminating in the Piazza di Spagna, which is known for its boat fountain and statue of the Virgin Mary.
Sadly, in all of this text, you've barely clipped the iceberg. There are numerous other neighborhoods worth a late night stroll, churches of high artistic and architectural value worth poring over, and many more piazzas in which to reflect on centuries and centuries of existence over a cup of coffee. Not only that, but we didn't even mention the Vatican! For more information and travel ideas when visiting Rome, take a look through our dedicated pages, utilize our trip planner or even watch some of our curated videos. You owe it to yourself to experience Rome at least once.