There are many good dive ops out of Kona but they all hit the same sites if you dive with them enough so really just find one you vibe with but the next time you come back it may be different as the guides jump around. We usually dive with a few different ops each time we go and mix in some shore diving. Below are my favorite ops depending on the area of the coast. Kahala Divers is my favorite op as they go to part of the coast that none of the boats out of Kona will go to so usually your are the only divers there. Blue Wilderness is run out of the General Store on Puako Road and can help with shore diving and renting anything you need.
Kona Area:
Pacific Rim Divers
Kona Honu
Aquatic Live Divers
Big Island Divers
Puako:
Blue Wilderness "small boat usually 6 divers or less"
Kahala:
Kahala Divers "My favorite"
Shore Diving:
Kahala:
Black Sand Beach at the Mauna Lani, Mahukona
Puako Road:
House 40, House 56, Catholic Church, End of the road
South of Kona:
Mile Marker 4, Two Step, Black Pebble Beach "My Favorite but not good on south swell"
Hilo:
Some good shore diving but the op we used is now retired so not sure who anymore.
Have fun, the more I dive the Big Island the more I like. We are lucky enough to get to go 2-3 times a year and we just got back last Friday and already miss it. Also we stay in a condo in Waikoloa Village which puts us kind of in the middle.