If you're planning a quick day in Athens before heading off elsewhere on the mainland or to an island, concentrate on the city's don't-miss classical sights, from the Acropolis to the National Archaeological Museum. In three days you can explore some key neighborhoods as well as monuments from the classical, Roman, and Byzantine periods; add two more days and you can take in some of the prettier suburbs, head to the beach, or check out new galleries.
Early in the morning, pay homage to Athens's most impressive monument, the Acropolis. Then descend through Anafiotika, the closest thing you'll find to an island village on the mainland. Explore the 19th-century quarter of Plaka, with its neoclassical houses, and stop for lunch at one of its many tavernas. Do a little bargaining with the merchants in the old Turkish bazaar around Monastiraki Square. Spend a couple of hours in the afternoon taking in the stunning collection of antiquities in the National Archaeological Museum (check to be sure it's open); then pass by Syntagma Square to watch the changing of the costumed Evzone guards in front of the Tomb of the Unknown Soldier. You can then window-shop or people-watch in the tony neighborhood of Kolonaki. Nearby, take the funicular up to Mt. Lycabettus for the sunset before enjoying a show at the Odeon of Herod Atticus, followed by dinner in the funky district of Psirri.
After a morning tour of the Acropolis, stop at the Acropolis Museum to view sculptures found on the site (note that the museum is in the process of moving to a new location below the Acropolis). Continue through Anafiotika and Plaka, making sure to stop at the Greek Folk Art Museum, the Roman Agora, and the Little Mitropolis church on the outskirts of the quarter. After a late lunch, detour to Hadrian's Arch and the Temple of Olympian Zeus, Athens's most important Roman monuments. In Syntagma Square, watch the changing of the Evzone guards, and then head to Kolonaki, followed by an ouzo on the slopes of Mt. Lycabettus at I Prasini Tenta, with its panorama of the Acropolis and the sea. Dine in a local taverna (Karavitis and Aphrodite are good choices), perhaps near the Panathenaic Stadium—in Pangrati, the National Gardens, or around the Temple of Zeus—which is lighted at night.
On Day 2, visit the cradle of democracy, the ancient Agora, site of Greece's best-preserved Doric temple, the Hephaistion. Explore the Monastiraki area, including the tiny Byzantine Kapnikarea Church, which stands in the middle of the street. In Monastiraki you can snack on the city's best souvlaki, then hop the metro to Piraeus to explore its neighborhoods and feast on fish in Mikrolimano harbor.
On the third day, start early for the National Archaeological Museum, breaking for lunch in one of the city's mezedopoleia. Swing through the city center, past the Old University complex, a vestige of King Otho's reign, to the Goulandris Cycladic and Greek Ancient Art Museum in Kolonaki or the more recent artifacts in the Benaki Museum down the street. Stroll through the National Garden. Complete the evening with ballet or pop music at the Odeon of Herod Atticus, a movie at a therina (open-air cinema), or, in winter, a concert at the Megaron Musikis/Athens Concert Hall.
Spend your first three days as detailed above. On the fourth, see the Byzantine Museum, which houses Christian art from the 4th to the 19th century; then visit either the Goulandris Cycladic and Greek Ancient Art Museum or the Benaki Museum—whichever you missed on Day 3. Another option is to check out the city's contemporary art scene at Technopolis or Athinais. In the evening, splurge on a meal at Aristera-Dexia; then dance the tsifteteli (the Greek version of a belly dance) to Asia Minor blues in a rembetika club, or, if it's summer, visit the coastal stretch where the bars stay open until dawn.
On the last day, elbow your way through the boisterous Central Market; then visit some of the many galleries that dot the area. Cut over to Kerameikos, Athens's ancient cemetery. After lunch in Thission, try a complete change from the urban pace: take the metro to the lovely suburb of Kifissia and view the grand homes from a horse-drawn carriage, shop, or relax in one of the many cafés. As an alternative, catch a bus near the Zappion hall entrance of the National Garden and head to one of the government-run beaches in Varkiza, Vouliagmeni, or Voula.