Hi Nick - I'll jump in to try and help since your post was directed at spearo's that live in the Azores and aparently no one here does. I have a couple of suggestions that might be helpful.
1). I did a dive in the Cape Verde Islands, several in fact, and it was very different from what I expected. My guess is the Azores is somewhat similar even being out in the Atlantic further and more north. The Azores has some big Blue Fin Tuna but way out of your league as a beginner. Most of the spearo's I read about in the Azores are Blue Water spearo's.
2) Go to some dive shops, boat marinas, etc., and ask around to see if any shallow water/reef divers might take you diving with them. Start slow and dive the shallow rocky reefs fr smaller reef fish.
3) get some inexpensive long plastic fins or even scuba type fins for shore diving. Get a good fitting mask simple J snorke (don't fall for the type that keeps water out with a stopper on top). A simple pole spear (forget speargun fr now) and get comfortable shore diving the shallow water. Oh, even though the Gulf Stream passes close to you - you will need a wetsuit - get an inexpensive 3mm wetsuit. Even a surfer wetsuit to start and of course a weight belt.
4). Many say take a course, dive with a buddy (good idea) but many of us (me too) started just like I stated. By ourselves exploring the shallows and slowly get comfortable to see if this really is what you want to do.
Good luck and come back to tell us what the waters are like, what type of shallow fish are found, etc. I've only heard about the giant Blue Fin Tuna spearing and would be great to hear what the shallow waters are like.
Most of all - be safe & have fun!!