Check-out Rome. I'd suggest visiting:
The Colosseum
Vatican
Sistine Chapel
St. Peter's Basilica/St. Peter's Square
Pantheon
Roman Forum
Borghese Gallery/Museum

I'm no world traveler, but I have been to Rome.
One thing we thought was cool about Rome is that some of these sights like the Colosseum and the Sistine Chapel are places I've heard about and seen pictures of since I was a young boy, especially the Colosseum. All the movies over the years in which the Colosseum played a part made it very cool to actually visit that sight and stroll around.

And definitely, absolutely buy tours for these sights. Purchase tickets through Trip Advisor. We did this for the Colosseum, the Roman Forum, the Vatican which included the Sistine Chapel and St. Peter's Basilica. We toured the Pantheon and the Borghese gallery on our own which worked out fine. The tour guides were very knowledgeable. I wondered if I'd get bored as these tours can be 2-5 hours long. Our Vatican tour was 5 hours, Colosseum/Roman Forum tour was 3 hours. Absolutely no hint of boredom. So much fascinating history that you'll never hear doing it on your own. I may never get back to Rome and feel the guided tours are money well spent. Our guide for the Vatican (Leonardo) spent the whole 5 hours talking about historical facts and stories about the people/painters like Michelangelo and Rafael. Do you know that the ceiling of the Sistine Chapel, painted by Michelangelo is a "fresco"? Fresco meaning it was painted by spreading wet concrete/stucco on the ceiling. Waiting long enough for it to set-up enough so he could draw his sketches on it and then paint it while it was still wet enough for the paint to settle into the damp stucco. He was commissioned by the Pope to do his painting. And (this is a BIG and) that was the 1st time he did a "fresco". A bit of a learning curve for him I'd expect. And imagine the pressure to get it right being requested by the Pope and all.

We stayed in Rome for 6 days and walked to all these sites. Really gave us a good insight into what daily life was like in downtown Rome. Rome is divided up into smaller "neighborhoods". Each neighborhood has it's own chapel. Step into some of those too. Just go in and sit for a little while. Beautiful small chapels in their own right.

Also, something else about Rome that we liked is that every year during Easter you will see TV coverage of what's going on at St. Peter's Basilica and Square. Yearly, that coverage reminds us of our fun trip to Rome and the sights we enjoyed there.