I'm not from Athens, but I know Athens as well as the locals. As a tourist, I'd recommend you go anywhere the sounds interesting to you. What is boring to me may not be boring to someone else. If you like history and looking at ancient things then you can pretty much go anywhere and enjoy what you see.
The places I would not recommend you go to (for your own safety) are: omonoia (very sketchy at night), metaxourgeio (very sketchy at night), plateia Amerikis (never go here as a tourist. It's very dangerous), exarheia, Acropoli (dangerous at night), plateia vathis (near monsatiraki/farmer's market), lofos filopappou (very dangerous at night. People are killed/robbed here at night). Pedion tou Areos (extremely dangerous at night. I'll never go here), Patision (the ghetto), and menidi (all of the gypsies live here. Cops will not go here). there are several other places too.
if you stick to the touristy areas then you'll mostly be fine.