If you absolutely don't have to drive while in Athens, save yourself the aggravation and use the public transportation instead.
The public transportation system is excellent, and besides the bus schedules being confusing, the metro, the tram, and the taxis offer an attractive and inexpensive alternative to driving in Athens.
I have driven in many major cities around the world and I have to say that driving in Athens makes the dreaded streets of Boston seem tame in comparison. Traffic jams are the norm in the center of Athens even during the late night hours, the roads are confusing if you have never traveled them, and the drivers in Athens have acquired a genetic "kill" instinct through the daily exposure to the Athenian streets.
There are many problems associated with driving and parking in within Athens city limits; too numerous to cover in one small web page. However, read on for to get a small impression.
Sitting patiently through heavy traffic to reach your destination does not mean that you will find a place to park somewhere within a reasonable walking distance. One of the main reasons of traffic congestion is the fact that many Greek drivers simply park illegally or double-park, and take their chances that while they take care of their business no large truck will try to go through the narrow street they block, and that no traffic policeman will pass by to give them a ticket or to confiscate their license plates.
Another major problem that makes driving in Athens very hard for foreigners is the erratic placement of the street signs that indicate the names of the streets. Such signs, which appear in both Greek and English, are nailed high on a the walls of the corner buildings and are often hidden by small trees, traffic lights, hanging plants and other such obstructions. Many times street signs are completely absent, or are placed only one one end of the street in a way that you have to look back to read them as you drive by. In any event, they are not legible until you are literarily underneath them, in which case it could be impossible to turn if you need to.
If you do miss your turn, don't assume that turing right, and then right again will bring you back to the same spot. Athens was not exactly built on a strict Hippodamian city plan with neat grids like, say, Manhattan. Instead it developed over centuries to a conglomerate of city blocks that define irregular narrow streets. As a result, many are defined as one-way streets since it would be impossible to fit two cars side by side through them. If you miss your turn then, you might find yourself driving for another hour around more very narrow streets.
But what makes driving in Athens most difficult for a foreigner is the erratic driving of others. While Greek drivers are attuned to the motorcycles squeezing between speeding cars as they pass each other, and to pedestrians saturating the streets from all directions, foreign drivers are filled with despair when they find themselves driving in such conditions.
Seemingly, there are no rules besides: a) get in front of the vehicle that's driving in front of you at all cost, b) fill every possible space on the road with a vehicle with no regard to traffic lanes, c) ignore any rules that do not seem convenient for your own personal comfort, and d) use the horn for any reason whatsoever.
If you do decide to drive around Athens despite my advice against it, remember that the lines which separate the lanes are purely decorative, and if there is a tiny space between the two cars in front of you (especially if they are stopped at a traffic light,) you absolutely must try to squeeze your car between them. You will be amazed at how many spaces you can squeeze your car through especially if you fold your mirrors.
Also remember that just because you turned your signal on to indicate that you want to change lanes, does not mean that anyone will give way. In fact, you will see that the great majority of Greek drivers (especially taxi drivers in Athens) have made a resolution to never let anyone get in front of them. Watch in amazement as they speed up to prevent you from changing lanes!
Above all, when driving in Athens, blend with the crowd: screech your tires when starting from a stopped position, and beep your horn in synchronicity with every traffic light as it changes from red to green. Heck, everyone else is doing it.
Seek the large avenues to move fast through the city. These are multi-lane, express avenues that are designed to move a lot of traffic long distances. They still have traffic lights, and they are still subject to gridlock, but they are your best bet for moving fast from one part of the city to the next.
The "beltway", Attiki Odos is a toll road (2.80 Euro for cars in 2011) that can take you across the north of the city and to the airport in a hurry. From it, several large avenues branch into and out of Athens. These include Leoforos Kifisou that connects Pireas with the Northern suburbs, and crosses the city of Athens to the west of the city the center. Similarly, to the east of the center of Athens, Leoforos Kifisias (that connects with Leoforos Sigrou) can move you from Attiki Odos to the north of the city with the coast of Attica (Faliro, Glyfada, and the rest of the Saronic coast). The southern coast of Athens is served by Leoforos Poseidonos that starts in Palaio Faliro and continues all the way to Sounion.
In the center of the city, on many roads the right lane is reserved for busses. You can enter the lane for a short period of time to make a right turn, but longer stretches might get you a traffic violation ticket. you will identify those lanes by the bus drawings on the pavement and by the double line that separates them from the other lanes.
Traffic in the center of Athens is restricted, partly to eliviate traffic jams, but mostly to minimize polution. The area is defined by a sign with a prominent "Δ". Cars with a license plate that ends in an odd number are allowed in the "daktylios" (or rign) on odd days, and cars with license plates that end in even number are allowed to drive in the center of Athens on even days. The only month that the restriction is not in effect is August when the city of Athens is almost devoid of cars (just about eveyrone is on vacation elsewhere).