Very few World Heritage Sites can claim to take up an entire country (only this one, really), and the Vatican City is an unusual country, to say the very least. You should go there just out of curiosity, even if you are not interested in anything else. The world´s smallest and least populated country, but one that issues its own coins and stamps, has a mercenary army of medieval Guardsmen, a border that is just a line drawn on the ground, and lots of other oddities in this last absolute monarchy in Europe (actually, a theocracy). Apart from all this, two top attractions await visitors: St.Peter´s Cathedral is an immensely big church with many side attractions (St.Peter´s Tomb, star marking the spot where Charlemagne was crowned Emperor in 800, the Pietà) and is usually very crowded (together with the no less magnificent St.Peter´s Square with its colonnades), especially on or around major church holidays. Probably the (even) better attraction is the Vatican Museums, an ensemble of museums, expositions, and collections that could take days to explore. Plus it contains a wonderful highlight - the Sistine Chapel, one of the world´s major masterpieces of art, in which papal elections still take place today. For this room alone, it´s certainly worth to stand in line for a couple of hours to get in. Often overlooked or bypassed, the Castel Sant´Angelo also merits a visit - very fascinating interior!
Even if you are not Christian or not a believer, you can´t fail to be impressed by these monuments.