This newly renovated hostel was an awesome start to my Azores trip. It was €9 by taxi from the airport. There's shared toilets and showers on each floor, which were always clean with plenty of hot water. There's an elevator to get your luggage upstairs. And the AC worked GREAT! (You're on a tropical island, so I suggest closing windows and using the air at night since Europeans don't have screens on their windows.)
The breakfast was plentiful for a hostel with bread, cheese, ham, jam, and butter. There was also fruit, yogurt, and cornflakes, coffee. There's a full kitchen if you want to cook and a mercado just around the corner to do your shopping. There's a pretty great and inexpensive burger place across the alley, so if you arrive hungry in the evening, that's an easy fix.
Gala, the house-mom/Herbergsmütti is super friendly and will answer all your questions with a warm smile! Her friendliness along with the guests made it a lot of fun! You'll easily meet others to do activities with, regardless of age.
Location is prime, so you can walk downhill to the city square, gate, and waterfront in 10 minutes. If you arrive by ferry, I'm guessing it's 20ish minute walk. Don't get a rental car until you're ready to go out of Ponta Delgada because parking is limited in the small streets, but you can park cheaply underground along the waterfront.
