Visit the Archaeological Museum of Piraeus. Take a forty minute metro (HSAP train) ride from the center of Athens to Piraeus. The train ride is mostly above ground, and it ends at the port. As you exit the station facing the port start walking to your left, following the coast until you find Charilaou Trikoupi street near the customs building and turn left towards Pasalimani harbor. From the metro station it would take a twenty minute walk to reach the museum. After the museum visit you can stroll around Pasalimani harbor towards Zea and watch for the remains of the ancient walls all around the coast.
Corinth can be reached via the "Proastiakos", the same metro line that connects the center of Athens to the Airport, but keep in mind that once you reach Corinth, you still need a way to get around there.
Sounio is about one hour drive from the center of Athens through the coastal road and can be done in one afternoon or morning. The major archaeological sites of Delphi, Mycenae, and Epidaurus would require an entire day for a visit since all three sites are about two to three hours away. These are extensive archaeological sites which require at least one hour for each visit.
If you only have time to visit one archaeological site outside of Athens, the site at Delphi would be the best choice since it is historically important, and the ruins are extensive. The drive there and the scenery from the site is spectacular, and the Museum of Delphi houses some of the most important artifacts from ancient Greece.
Mycenae and Epidaurus could be visited together during a long day trip. Leaving Athens early in the morning would be best to visit both sites. Near Mycenae, one could also visit Tyrins, Argos, and Nayplion which is a beautiful town with nice beaches near Tolo.
Athens Zoo is a pleasant place to spend the better part of a day, especially if you travel with children.
Of course you can choose one of the numerous beaches around Attica. They are not of the quality you find at the islands, but many are more than pleasant to spend a day at. Read our guide to beaches of Athens...
You can take a daily trip from Piraeus to the islands of Aegina (don't miss the temple of Aphaia), Poros, Hydra, and Spetses. You can take a slow ferry, but a "flying dolphin" (catamaran) will make the trip in a fraction of the time, so you can be back to Athens for dinner. All of these islands are excellent day-long destinations, but if you have the time you might want to schedule a day or two at Hydra island.
There are several companies that offer daily cruises to the Saronic gulf islands. They usually leave around 7:00 AM from Faliro and sail to Aegina, Poros, Hydra, and then they return to Athens early in the evening. Some of these small cruise boats make a stop at Spetses after Hydra.
The service varies from boat to boat, but it is usually a far cry from the cruises that sail the Aegean islands for three to seven days. You can expect to be packed among many many passengers with nothing to do besides drinking (something the cruise lines really want you to do). They are a lot slower than the regular ferries and catamarans, but they offer you the choice of seeing several islands without the hustle of changing boats.