The Athens International Airport (Eleftherios Venizelos) was opened in time for the 2004 Olympic Games. It is clean, modern and up to date.
The cheapest way to get to the airport is by bus. There are several buses:
X95 - To Syntagma Square (runs 24 hours a day)
X94 - To Pentagon Metro Station
X96 - To Piraeus (ferry port)
Tickets cost just slighty over €3 per person if you're taking the bus.
At the moment, the metro/train line (blue line) to the airport is closed till close to the end of 2008. When it re-opens, it costs €6 for 1 person or €10 for 2 persons to get into Athens. You'll likely have to transfer to either Syntagma or Monastiraki to get to your hotel. The cost of transfer is included in your original ticket. The metro is very clean and efficient. There is plenty of room for luggage and you won't have to worry about getting lost because the metro maps are displayed inside the cars and also provided in English.
As for now, taking the bus seems to be the best option although I've always used the blue line metro to get to the airport. This is just a temporary option till the rail line re-opens. If you have a late night or early morning flight, the X95 bus is your best option as it's 24 hour service.