Greece is a country that is famous for many pilgrimage sites. (Sorry 10 were limiting ...)
The most famous and available to everyone are Patmos, Tinos, Meteora and St. Paul steps.
Mt. Athos is for men only and a permission for entry must be requested well in advance. However during the tourist season from Ouranpolis there are day cruise trip to see the monastery from the sea.
Ossios Loukas monastery not too far from Distomo and Delphi is a UNESCO world Heritage site with the beautiful mosaics. Nea Moni in the island of Chios is another UNESCO world heritage site with mosaics.
Tinos is often referred as the Lourdes of the greek orthodox, just facing Mykonos island with a short 30 minutes crossing.
St. Spyridon in Kerkyra, Corfù island. St. Dionysios of Zakynthos in the main island town.
Panagia Ekatontapiliani in Paros where St. Helena stopped on the way to the Holyland.
Panagia Hozioviotissa in Amorgos (in the picture).
If you are only travelling to Athens, try to visit the monastery of St. Nektarios in Aegina island 40 minutes from Piraeus with Flying dolphin. St. Ephrem of Nea Makri Monastery in Nea Makri just after Rafina port.
St. Andrew cathedral in Patra and Nafpaktos just on the other side of the bay famous for the battle of Lepanto.
St. Micheal monastery in Panormitis, island of Symi. St. Michael monastery in Mantamados in the island of Lesbos.
Byzantine churches in Mystras next to Sparta.
Byzantine churches in Thessaloniki.
Every island and every major town has his own saint or miraculous icon of the Mother of God. Just ask at the reception of your accomodation for more details.