Most visitors to Athens come to Exarhia only to visit the National Archaeological Museum, a tour of which can easily fill a half-day. But even if you're not particularly interested in Exarhia's counterculture, it's a convenient place for a lunch break, with a big variety of affordable restaurants, from student-casual to just the right amount of elegant. As for Omonia, you don't have to plan to come here, precisely, as so many streets and metro lines lead here, but when you do, take a coffee break on Plateia Omonias to appreciate the daytime bustle and great cross-section of people.
If you want to dig deeper in Exarhia, take a stroll down side streets to look at the most modern of arts: fantastical murals and pointed graffiti, some whimsical, but most with a serious message. And on any given day, there's usually some kind of political activity on Plateia Exarhion – banners at the very least, and possibly a demonstration. The area is also known for its hole-in-the-wall collectors' shops, particularly records, so if you're a vinyl freak, allow some time for crate digging too.
Be sure to circle back to Exarhia in the evening if you like casual drinking holes, and especially if you have an interest in classic rembetika (the last of the great late-night venues is here) or the interpretation by younger musicians that has become so popular in recent years. Rather than eating dinner in one restaurant, you can bar-hop a bit, following your ears to good music and eating mezedhes (snacks) with every round of drinks.