Went to Athens for about a week in September. We stayed right outside of the Plaka area near Monastiraki Square. We got around the whole trip between walking and using their Metro. Their Metro was really nice and super easy to use, I'm not a big traveler, but I thought it was easy. You can actually take the Metro right from the airport to that area for a couple Euro if you don't have transit set up all ready, we actually did that there and back when we took a mid-trip trip to the Islands for a couple days.
As for what to do, we had a really great walking tour that hit the Acropolis, Ancient Agora, Royal Gardens in about a half day. We enjoy the small (10-15 person) guided tours over just walking around that stuff by ourselves. The Acropolis Museum wasn't huge, but was a really nicely done and had all of the preserved statues they have left from the actual acropolis site. It wasn't a part of our tour, but we went the next day and it was nice to see the sight first, then see the artwork from it. Besides that we did a lot of walking around the Plaka area and just finding street cafe's to eat at and browsing.
We really enjoyed our hotel, 360 Degrees Pop Art, small but nicely appointed rooms, and location was really good right of the Square and near Metro.
ETA: We aren't really nightlife people, so we were out late ever, but when we were out (days/evening), we felt pretty comfortable walking around those areas. I heard so many warnings about Athens being dangerous, but in the area we were in, we felt safe the whole time.