Athens and Santorini
I have to recommend you several places to eat like Sea Side at Perivolos or Perivoli Restaurant at Exo Gonia. You can have even a cooking class apart of your meal to the Restaurant Selene at Pyrgos, where you'll find it's owner, the best local goods ambassador of the island. There are many other places to eat but i am not sure if they'll be open in April.
Do not forget to drink Vinsanto a local sweet wine, to visit the unique vineyards of the island and why not one of the many wine estates. Also don't miss the white eggplant, the fava beans with capers, and the Santorini cherry tomatoes, all local products of this blessed island.