Oh God! Driving is a really difficult thing in Athens but it can be really fun, if u have a lot of patience. First of all: if u want to visit central areas, get a subway map. You can leave your car near a subway st. and then continue your trip down town. Note that not all subway stations have parking facilities. You can use public garages or hotel garages for your accomodation. If you are stuck in the centre, Hilton Hotel has garage where you can leave your car, even if you are not a client. You can go to Evangelismos St. by foot ( 2 min. ). Also Titania Hotel at Panepistimiu Av. is a choice if u are stuck in the downtown traffic. Things with traffic are dramatically better during Saturday and Sunday. Generally you can leave your car in Ethniki Amina St. in the North and in Sigru Fix St. ( with garage facility) at the South. Prefer Attiki Odos, Immitos Ring and Sigru Av. for reaching the centre.
Old Airport Area hotels are a good choice for staying and parking in the road ( Glifada, Vula, Vuliagmeni. Elliniko ). Note that some small quantity of areas have parking tickets for the road.
As for suggestions i would suggest visiting Sounion Temple by the car, Varkiza coast for food and leisure and take a small trip by car to Epidavros Theatre to the South ( not a short trip ). Also if you dont want to go to the basics try a 1 - day trip to Aegina and Angistri island by your own. It is a 60 min. trip from Pireas port and you can also take a slower ferry to visit Aegina if you want to use your car too.