I would suggest looking at the
ABD itinerary for Athens to see what is done. We saw most of the major sites in Athens and I came in early to see other sites that weren't on their list. Most of the major sites in Athens are within easy walking distance of the Acropolis. I used the Frommer's map for Athens to plan my additional activities. Your ticket to the Acropolis should also get you into the Agora(below the Acropolis) that same day for free. I visited several museums(Benaki, New Acropolis Museum, National Archaeological Museum). Temple of Olympian Zeus and Hadrian's Arch are across the street from the Acropolis. Hadrian's Arch is on the street and the temple you can see through the gate behind it, so if you don't want to pay to go in you don't have to. The metro stations have actual artifacts in them, so take in the view before you get on the train. You also have to validate your metro ticket at the slim yellow or white machines that are at the long entrance hall to the tracks. Don't forget to do it because they sometimes have workers at the other stations who ask you for your ticket to see if you validated and paid the fare. I didn't do Filopappos Hill or Mt. Lycabettus because I didn't have time, but you may want to visit them. They take a little more to get to. Parliament and the Tomb of the Unknown Soldier in Syntagma Square are right in town by the metro. They have the changing of the guard, which was fun to watch. If you want authentic handmade Greek sandals, visit the Poet Sandalmaker(his son makes sandals now) in Monastiraki Square(behind the Agora)--you should be able to Google him or find him in tripadvisor. They only take cash(euros), but the sandals are fitted to your feet and are soooo comfortable. Take your patience, the process can take about 1-2 hours depending on whether a cruise ship is in port. Also, check to see if your hotel has a free shuttle to Syntagma Square. The one ABD stayed at did and it made it very easy to get to shopping, restaurants, and the sites.
Santorini is all about the caldera and the volcano. When I did the ABD in 2012, we hiked one of the volcanoes, went to the Volcan Winery, Akrotiri and some other places. If you want to visit a beach, Kamari Beach was great although it was rocky(not like beaches at home).
The food in Greece is awesome! I now see what all the fuss is about the Mediterranean diet. When I did my initial research, I remember reading that Greek restaurants charge a bread service fee--and no you can't refuse the charge by not getting bread. It's basically a fee for using the table. Unlike in the U.S., in Greece you won't be rushed through your meal. You can sit at the table for as long as you want to.
I also find Tripadvisor extremely useful when planning trips. I search the things to do and decide which I'm interested in. Good luck on your trip and I hope this helps.