If there's something that you will not miss in Açores is the green landscapes, even in the city you can find wonderful gardens full of vegetation, huge trees and all kind of flowers. In Ponta Delgada my two favorite ones are Jardím José do Canto (2€ to get in) and Jardim António Borges.
Jardim José do Canto. It is truely one of the richest and largest gardens on the island which has been visited by many famous personalities. It includes nice architecture such as The Chapel of Saint Anne from the XVII Century.
Jardim António Borges. The garden is named after an Azorian politic interested in botanic. He introduced exotic kinds of trees and plants to his garden, becoming a wonderful Romantic garden in the middle of the city.