Seriously...you could spend a month or two here and still not see everything.

Nav:

There are so many sites so you have to prioritize what's important to you in the time you have.

The Pantheon
The Roman Forum...aka. Main Street
The Mamertinum...Where the Apostle Peter was held before execution.
The Spanish Steps, because the Spanish Embassy used to be at the top.
Trevi Fountain
Legend holds if you throw a coin in, you'll return to Rome one day.
Flavian Palace - Home of Emperors...
But extensive ruins today.

The Victor Emmanuel II National Monument (Vittoriano or Altare della Patria). A monument to the first king of a unified Italy.

Nosh:

Literally eat anywhere. You won't be disappointed (unless, maybe, you're actually Italian).

Lasagna
Pizza
Don't forget gelato!

Nip:

Drink the house wine wherever you eat. Red, white, doesn't matter. Just order it and enjoy. It's tasty and cheap...what else do you want?!