This is almost entirely a nightlife district, unless you're into street art, in which case a daylight stroll is rewarding (fold in a walk through the old gasworks too). The only other daytime sights are the Kerameikos site – go early in the morning, because there's no shade – and the two Benaki outposts, one museum for contemporary art, and one for Islamic art. These require a little planning, as they have more limited hours and, in the contemporary space, rotating exhibits that may or may not be of interest.
Central Gazi, around the Kerameikos metro stop, is a virtual ghost town by day. Some cafes and restaurants are open, but there is very little street life; the area just feels like it's biding its time till sundown. So for a break after the Benaki Museum Pireos Annexe, you could eat at its very nice (if somewhat pricey) cafe, or head across the road to livelier Petralona.
When you do come back in the evening, where you settle in depends a lot on your taste in bars. Stay in Gazi for bigger clubs and louder lounges; check the schedule at Technopolis in advance, in case a band or DJ you like is playing. If you prefer a slightly quieter spot, head east of Iera Odos. But even if you're not into the high-volume Gazi scene, it's fun to at least stroll through the main square, where people meet to preen and decide which place to hit first.