If you’re visiting from afar then you’ll fly into the Athens International Airport – Eleftherios Venizelos Airport (ATH). To reach the city center you can take an airport express bus or the Metro Line 3 (Blue Line). Metro trains leave every half hour and run 7 days a week from 6:30 am – 11:30 pm and lasts around 40 minutes to Syntagma station (center). Express buses run 24 hours and connect to both the center and to the port (Piraeus), among other options. Check which route is best for you.
Another common way to arrive in Athens is via ferry. Whether it be to and from the Greek islands or internationally (mainly Turkey and Italy). This guide mainly deals with domestic ferry travel but has good information that can be applied to international ferry travel as well.
Athens isn’t super well connected via train but you can find one from Thessaloniki (which connects from various other Balkan countries) or a few more suburban trains. For more info, everything train related goes through this guy.
Bussing is always a cheap and easy option if you’re coming from Istanbul, Thessaloniki or other nearby destinations. Visit KTEL for direct bookings and info. I also use Balkan Viator, Bus Ticket 4 Me and Get By Bus as other resources in that area.
You could flashpack in 2-3 days but 4-6 would be ideal to explore further and take some day trips