I really enjoy Athens and have lived there for several summers over the past few years. Traveling around Athens is very easy because their metro and bus system was revamped for the olympics. You cannot go wrong with a day pass. Taxis are generally cheap and the drivers usually don't take you for a ride, since they've been more regulated.
If you're looking for touristy crap to buy for people back home, the Plaka is the place to go. It is in the area next to the Akropolis and you can reach it on the Green line metro, stopping at Monastiraki. Just to the west of the station is a restaurant called Savvas (Σαββας). There you can get .5L beers at a good price, and if you're looking for a quick lunch you can walk around into one of the alleys near the store front and order a gyro that costs 3.5 euros that is absolutely delicious. I would suggest the pork. Moreover if you sit down and eat there, get the gyro dinner plate. It is a huge pile of delicious meat atop a pita. You'll not go wrong with that.
In that area you can see the Athenian Agora at the bast of the Akropolis. I excavated there for a couple of seasons and it is a really cool place. Their museum is short and sweet AND air conditioned. The Areopagus, the hill right next to the entrance up to the Akropolis is a fantastic place to go at night to hang out and see the city. I'd suggest getting a bottle of wine (or two) and hanging out there for a bit. Also one of the places I like to stay (Hotel Thession) is very close by. It is right off of Apostolou Pavlou, which is a walking street. Rooms there are generally 60 euros a night. It's nothing super special, but it is air conditioned and you can wake up and see the akropolis right from your pillow. If they're full, there is another hotel a couple streets back that also would suffice at comparable rates.
Kolonaki is a neighborhood to the west that has some excellent restaurants and is at the base of Lykavittos Hill, where there is a stadium and a breathtaking view of the whole city. That is accessible from the Blue line, exiting at Evangelismos. That part of town tends to be more expensive, but they do have a park where they show films out doors after sunset. I would suggest eating at the Fagopotion ( Φαγοποτιον), which means something like the :"eat-drinkery". The food there is excellent and the people are nice.
Depending on how adventurous you are, you can walk around Exarchia, which is the neighborhood just to the north of Kolonaki on the other side of Lykavittos. The university students tend to live and hang around there. Also in Exarchia is the National Archaeological Museum. This is THE museum to visit for ancient Greek (and Roman) art. Be wary when walking around the National Museum's grounds. Between it and the University (called, colloquially, Panepistemio, which means University) is a street full of drug addicts (called Toxomanes, or "chemical-maniacs"). The police, after the military junta was ousted, cannot enter a university campus, so the drug addicts hang out there as they can run onto campus (or try) to avoid the police.
I'd also check out the Benaki Museum on Vasilisis Sophias Street. It is a privately owned museum and has some amazing byzantine art. The Cycladic museum, the Goulandris, is also a private museum that has some very beautiful art from the Islands. Additionally the Numismatic museum in itself is alright (I can't get into coins), but it is also housed in Heinrich Schlieman's home. Schlieman was the man who first excavated Troy, and the house is absolutely beautiful. Check out the new Akropolis museum too. It looks like a parking garage, but is very well put together with the artifacts found on the akropolis. As for visiting the Akropolis itself, get there early in the morning before the cruise ships dump off their passengers. It is nice and quiet and you can see things without having to worry about elbowing your way through crowds.
Walking around the Piraeus is also kind of nice. If you write to the Zea Harbour project, run by the Danish School of Archaeology, they might be included to give you a little tour of what they have done so far, perhaps show you a video.
Generally, you cannot go wrong with food, provided that you get away from the touristy places that will overcharge you. If you can speak Greek, or even a few phrases, the people will LOVE you. You'll see newspaper stands all around the city too. The stand is called a "Periptero" and in addition to newspapers, they also sell bottled water (which is dirt cheap) and tall boys of beer (which is also cheap). There are no open container laws, save for having them in cars, so one need not worry about enjoying a brew in the open.
I've not been to Mykonos or Santorini. If you're taking the boats, be careful because this time of the year the winds can be terrible and you may get stuck on the islands for a while. One can fly relatively cheaply with Olympic or Aegean air, but you get what you pay for.
If you are flexible enough, try to visit Crete. It's one of my most favorite places in the world. Hania is a beautiful Venetian port city that is quite romantic. Delphi is a good day trip from Athens and you can take a bus pretty easily there. Cape Sunion is also a good day trip and the beaches there are pristine. If you're in to it, eat lots of calamari because it is fresh and delicious. Stay away from fish because it is super expensive, as the Greeks have to import it since they've over fished the area.
If you have any more questions, I'd be happy to answer them. Just shoot me a message!