Kolonaki is in Athens near Lykavittos Hill.
The district's borders are not very sharply defined; it covers the south and southeast slopes of Lykavettos Hill north of Vassilisis Sofias Avenue. Kolonaki is the posh area of central Athens. Traditionally the home of the in-town rich, it's the location of a number of foreign embassies and several prominent archaeological schools, including The American School  and The British School . It also has the city's greatest concentration of trendy fashion boutiques, and many, mostly upscale, cafes, bars, and restaurants.
The Metro stop for Kolonaki is Evangelismos - upon exiting the station, turn north and walk uphill. Walking to Kolonaki takes about twenty minutes from Syntagma and a half hour from Plaka, though much of the walk is along congested, difficult to cross streets.
Kolonaki used to be one of the nicest areas of Athens to walk around in, but in recent years the auto traffic on its narrow lanes has gotten congested beyond belief, especially on its lower slopes, making crossing the street anywhere in the district an unpleasant and even dangerous experience. Still, if you can choose a relatively low traffic period, a walk through the district might be worthwhile. Start off in Plateia Kolonaki and Plateia Dexameni, the main shopping street of Patriarchou Ioakim, and investigate the parallel streets up from it, such as Haritos, with its mixture of old neoclassical houses, upscale modern flats, and art galleries. Speusippou, which turns into quiet Souidias, is the home of the Gennadius Library. Further up the streets, climb steeply to the Lycavettos funicular station. If you're ambitious and energetic, you can climb to the top streets which to your left circle the rim of Lykavettos, which are successively Aristippou, then Il. Rogkaku. On this path, from the cross streets and some spaces where there are no buildings, you can glimpse spectacular views of Athens, some of them including the Acropolis. If you keep going around you'll reach Sarantapichou Street, but by this time you've left Kolonaki and are looking down on the student district of Exarcheia.
Kolonaki offers the usual range of shops you would expect from an upscale neighborhood; art galleries, hip clothing boutiques, and antique stores are thick on the ground. Since the area is small, and the boutiques especially tend to come and go, rather than make a list of promising shops the traveler is interested in, you may find it preferable to just walk around and discover the stores on your own, especially around Plateia Kolonaki (Kolonaki Sq) and the small streets north of it, including Skoufa, Anagnostopoulou, and the pedestrianized Tsakaloff. Walking along Patriarchou Ioakim and Haritos Sts and their cross streets, there are plenty of stores to wander in and out of.
Kolonaki Square (Plateia Kolonaki; its official name is Plateia Filikis Etaireias, but no one ever uses it) is bordered with cafes whose customers tend to be drawn from the class of mature Athenian movers and shakers for whom the neighborhood is the traditional in-town home. Most of these cafes serve desserts and/or light meals as well as drinks, and most of them are expensive. They tend to be liveliest late at night. One of the best established, and most prestigious, is Lykovrisi. Not many foreigners show up in these cafes, but the visitor may find they make for interesting people-watching.