Athens has the most famous skyline in the Western world and one of the least pretty cities around it. The Parthenon floats on its rock above a sea of white concrete apartment blocks, and that contrast is the city: Athens invented democracy and theater in the 6th century BC, built the Parthenon and most of classical philosophy in the 5th and 4th, then spent two thousand years as a provincial town under Romans, Byzantines, Franks and, from 1458, Ottomans. When it became the capital of a new Greek kingdom in 1834, it had a few thousand people. Today it has nearly four million, most of whom arrived in the last hundred years.
Don't let the concrete fool you. Athens is warm, loud, generous and much more fun than its reputation: neighborhoods full of tavernas and cafรฉs, markets that still feed the city, rooftop bars facing the Acropolis and a nightlife that starts around midnight. The ancient sites are walkable from each other, and the sea is half an hour away. Five days lets you see the great monuments and museums without rushing, and still have time for a long lunch and a day on an island.
This plan keeps each day in one part of the city. Day one is the Acropolis, its museum and the Ancient Agora, day two the old town, from the market to Plaka, day three Syntagma, the Olympic stadium and Lycabettus, day four the National Archaeological Museum, Exarchia and an evening by the sea, and day five a boat trip to the island of Hydra. If you have longer, the best trips out of the city come after day five, Delphi and Cape Sounion among them. Athens is also the best place to start exploring Greece: every ferry and almost every bus leaves from here.
Start on a Friday if you can. That way you get the Acropolis Museum's late opening on day one, the market and the Saturday street market on day two, the full changing of the guard on Sunday morning, the National Archaeological Museum open from 8am on Monday, and day five on a weekday ferry.