Of course, when you go to Athens, you go to the Acropolis. Let me tell you -- it is worth the walk! You can take the Metro to the base (the Akropoli stop). It's quite a hike up the hill, but once you get there and are standing on that rock, it's quite a feeling. And what a view! There are several ruins on the Acropolis. You buy a ticket to be able to see the top Athens attractions. There is a path that goes back down and into Ancient Agora. There are also other paths that go to other sites.
In addition to the Acropolis, I would recommend the Temple of Zeus. There's not much left, but it is still impressive. It's so huge that it's unbelievable to imagine how they built it. Also, a trip up Lycavitos Hill is worth it. It's the highest point in Athens, and the views are spectacular. There’s an old church up there and a restaurant for the tourists. You take a cable car to get to it. You can take a taxi to the base or walk through the hilly neighborhood to get to it.
One funny thing about Athens: there are many dogs that just roam around loose. They are even at the ruins sites! We thought it was funny that we weren't allowed to touch the ruins, but they let the dogs just sleep there or even pee on them! I guess there’s no animal control there.
One more thing: I would highly recommend a visit in the off-season. It was so wonderful to not have to share these sites with 3000 other people. The weather was still very pleasant, in the 60's or 70's, nice, but not too hot. And at the most, there were maybe 100 people around. And that was at the Acropolis. At the other sites, there was just a handful of people -- it was wonderful. We got many pictures without anyone in them, and we could stay and enjoy the sites as long as we wanted.