Vatican City merits most of one day, especially if you’re doing a guided tour of the Vatican Museums and St. Peter’s Basilica in one go. I do recommend getting a spot on a guided tour of Vatican City, partly because the museums are so vast it’s impossible to know what’s important to look at and what’s not (unless you’re a history buff or an art major) and partly because knowing the context of what you’re looking at makes it infinitely more meaningful.
A guided tour typically starts by 9 or 10 in the morning and can last 4-6 hours, usually with a break for lunch between a swing through the museum and a tour of the Basilica, and most guides arrange it so you’ll meet them in Vatican City. Be aware that St. Peter’s has a strict dress code (many churches in Italy do, but few enforce it to the degree that St. Peter’s does) – no exposed shoulders, knees, or midriffs are allowed. I’ve seen people walking through this most dignified cathedral wearing disposible paper clothing over their shorts and tank-tops after being forced to buy the paper coveralls from a nearby vendor at a ridiculous price. Don’t get caught in this situation.