Visitors usually come to Kolonaki first for its superb museums. Given the size of both the Benaki Museum of Greek Culture and the Byzantine & Christian Museum, it's hard to recommend visiting both in a single day. The Museum of Cycladic Art is smaller, and a manageable add-on to the two biggies.
Whichever you visit, it's nice to fuel up with coffee first on the neighbourhood's main square, Plateia Kolonakiou. Lunch at the Benaki Museum is lovely, and then you're set for a little shopping at Kolonaki's many boutiques. If this appeals, be sure to note typical Athens shop hours, with 'late' opening (to 8pm or so) only on Tuesdays and Thursdays; other days, most places close by 5pm or 6pm.
As in most Athens neighbourhoods, there's a hill to climb. Lykavittos is taller than most, but it offers a funicular as a shortcut. The shady footpath up is pleasant, too, when the weather's not too hot. The view at night over the whole glittering cityscape is impressive, and there's a bar-cafe up here to enjoy.