So to and from the airport and the port are the two big reasons to use the metro that I can see. By the way you can use the Syntagma Sq. Tram on Queen Amalias Ave. to get to the seas shore and beaches up to Glyfada because the metro doesn't go.
Since central Athens is a walkable city and in the historic center its often a delight to walk in, so you likely wont use the metro that much in Athens center. Still all things considered and especially if your dogs are barking (feet hurt) the metro can come in handy. Many should pace themselves in the intense (but dry) heat. Wear a soft hat too something you can get wet later in the sea! Shades manditory!
You can plan your walking tours with the metro in mind like I already did for you and Pausanius did for all us travel writers. You'll enjoy some major sightseeing with your outbound tour on foot and the inbound by metro or verse visa.
Left: Using the Athens Walking tour Map III for example: a walk from Syntagma Sq to Thession station or brand spanking new Keramikos Station (06/07) would enable you to see a lot of old Athens including the Agora and Ancient Cemetery!
Then satiated after a repast of grilled eggplant, goat cheese and chilled rose wine, or an ouzo and octopi this skaras meze (octopus to be grilled hanging photo right, click to see larger).
After your meal, you could simply hop on the metro in Thession and return back to Syntagma Sq. and your centrally located Athens hotel. But this requires you change lines once, from green to blue at Monastiraki to reach Syntagma Sq. If you were staying in the Exarhia or Omonia Area you would be all set and wouldn't need to change lines. Besides Thession and Psirri, there are restaurants in the Monastiraki and adjacent Plaka area you should consider too.