TAXI with Welcom Pickups:
A taxi for 6 people + luggage with Welcome Pickups: €85
2 x Regular not pre-booked TAXIS: 2 x €40
€ 40 is the taxi official flat rate to Athens center. (4 passengers max)
Taxi will drops you off in front of your accommodation.
For Athens center The journey is a little faster than by public transport (10 to 15 minutes)
METRO (Line 3)
6 full-price metro tickets to the center of Athens: €54
The journey to Syntagma Square takes 38 minutes. There is a metro every 36 minutes.
You will probably need to walk (or take another metro or bus or taxi) once you arrive at the correct metro station.
It all depends on where your final destination is. I can advise you on this point if you specify where your final destination is.
The metro is a pickpockets playground, especially on the section you are going to take.
That being said, you just have to take some common sense precautions.
Pickpockets' favorite moments is mainly when you are about to get off the metro train. If someone tells you the door is stuck and asks you to go to another door, it's a pickpocket with a 99% chance. Arriving in front of the escalators when there are quite a few people is also a risky moment.
6 X95 express bus tickets to Syntagma Square: €33
Same remark as for the metro regarding the need to take another transport or walk once you arrive at Syntagma Square.
A bus every 15 to 20 minutes, journey time 40 to 50 minutes depending on traffic. No or almost no pickpockets on the bus.
There is a 3-Day Tourist Pass at €20 which includes 3 days of public transport, metro, bus, trolleybus, tram) plus a round trip Airport ↔ Athens by metro or Express bus.
The suburban train "Proastiakos" which leaves from the airport does not go to the Athens center where your destination will probably be.