You don't need to plan to go to Syntagma – you'll naturally pass through this hub. If it's near the top of the hour, stop to see the changing of the guard in front of the Tomb of the Unknown Soldier. The area can feel a bit bland on the main streets, but side streets reveal surprises, and the square itself is a treat in the evenings, when a cross-section of Athens gathers on the wide stairs, in front of the multicoloured fountain, even at the outdoor seats at McDonald's.
You'll also want to come to this area in the evening for a drink or a few. Some of the city's best bars are here, tucked away on pedestrian streets or down the end of shopping arcades. The scene naturally bleeds into Monastiraki to the west.
As for Plaka, it's naturally a draw, and it's easy to spend the better part of the day looping around its curving streets. There are few must-do sights – the National Historical Museum and the Jewish Museum call for the most time. Otherwise, there are just a lot of smaller quirkier things to discover – and this includes the centuries-old churches scattered all around. They don't have set opening hours (and there are many more than we list), so just pop in whenever you see one open. Likewise, if you see any interesting set of stairs, follow them – you might get lost, but that's part of the pleasure of this neighbourhood.