In Athens, the public transportation system is pretty good. There's a line from the port into town and another one from the airport that takes you right to Syntagma Square, the city's central square. No need for a taxi, in my view.
In Santorini, right, the town ebbs and flows with tourists from the ships that call there. We stayed three nights at a hotel called the Homeric Poem, had a view of the caldera, and its sunsets, to die for, and had a great time exploring the town before and after the cruise ships.
You might want to consider renting a scooter to explore Santorini. That's fun. Also, the shops in Santorini, and everywhere else, are filled with almost identical tourist junk. Except for one place. Try to find a shop called the Handy Jewelry Studio on kind of a back street behind the old covent right in the middle of Santorini, kind of to the left on the main pedestrian walkway as you come up from the ships. Beautiful stuff at reasonable prices and a very interesting and personable artist.
If you're a glutton for punishment, you can read a good deal more about this on my blog, The California Files, here:
The California Files: Greece, Part Four