If you plan to visit several European cities, make Athens your last stop.
How can you go to Athens and not visit the Acropolis and the Parthenon? The answer is that you cannot.
Once you have visited the Acropolis in Athens, the appeal of other attractions diminishes.

When you get to Athens, plan your visit to the Acropolis, if possible, on Sunday.
Before you visit the "sacred rock", start with the New Acropolis museum. The outside is incredible. You walk over glass panels through which you can see excavated ruins. In the museum, start again from the top floor and after watching the small video and learn about the history of the site start walking down.

On Sundays all attractions open to the public at 8.00 am. I would say though if you can get here early, it will be well worth your time. Be there when it opens. You'll have time to wander around and experience the amazing site with only a few people around, not enough to ruin or get in the way of your experience. You can spend hours here. The view of the city of Athens from here is incredible. From this vantage point you can also see the location of many of the other sites worth visiting while in Athens. Hoards of tourists arrive around 10:30-11:00 and you will have trouble seeing things, much less getting good pictures, if you go during the middle of the day when everyone else does.

The Acropolis is amazing. There really isn't a whole lot more to be said about how stunning it is than what's already been said a thousand times over. When you arrive on the top of the hill and you see Parthenon so close, you really are speechless. There's something so special in the air just for having the chance to be there, and look at, and walk at the very same place, thousands and thousands of people have lived, all these centuries. It is a masterpiece, and you definitely realize that when you see it.

A few words of advice: bring water bottles and wear flat shoes for the walking.