As with all the archaeological sites, it's nice to visit the large Ancient Agora early in the day; here, you'll probably spot birds and critters if you go first thing. Unlike the other sites, though, it does have some shade. Allow about an hour and a half.
The nearby Roman Agora is far smaller; the key attraction is the Tower of the Winds, most of which you can see from outside the fence (there are signs that explain what you're seeing). But if you're interested in the Ottoman period in Athens, go inside to see the interior of the tower (which Sufis used as a meeting place for a while) and the restored mosque on the site. Also don't miss the gate of the Sufi madrasah across the road.
After your key sightseeing is done, there's plenty more to browse, and you can shop and bar-hop all over the area. It's nice to head to Varvakios Agora, the central market, around lunchtime, and then you can eat at one of the deli-restaurants on Evripidou or a more traditional taverna in the market proper. If you're especially food-focused, though, try getting to the market around 7am when deliveries are happening and it's the most bustling; you could even have breakfast with the workers at one of the tavernas.
For drinking destinations, try to make it to at least one rooftop bar, as the views of the Acropolis are great from here. And, since Monastiraki is basically souvlaki central, plan to have a sandwich at one of the classic spots here.