Pangrati is my neighborhood. Outside of the first Olympic stadium, there are few tourist attractions. Most of this area's energy just comes from Athenians going about their daily lives. You'll find grocery and clothing stores—slices of real Greek life.
If you have time to wander, Pangrati is the perfect place to feel like a local. Keep in mind that tourists don't normally venture here, so you'll run into non-English speakers, as well as some eccentric locals. As with all neighborhoods in Athens, it's easy to get lost, but that's part of the fun. Don't miss the liaki agora, or weekly street markets. There are several in Pangrati. They may be hard to find, but they're lots of fun to explore.