There are several good dive shops on the Big Island, and in Kona a good one to check out is Bluewater Hunter (Rob White). He was helpful to us when we dove Kona side.
I've tried a few spots there. Being from Hawaii, we tend to look for places off the beaten track. But I've found some good places in common spots. If you are staying in Kailua-Kona there are lots of good places on Alii Drive, which is where many of the hotels are. North of the airport there are also some good places where the resorts are at. I would recommend you dive near the hotels/resorts because these are generally safer (someone is more likely to see you and you'll be near facilities). Or get a snorkel map - many of those places are great even though they are pretty obvious and get a lot of divers.
Remember, it's required by law to use a dive flag if you are diving. If not, they can take all your gear and fine you and it's not uncommon. It's also not safe to dive without the proper gear. I don't know if you can rent a float, but we have Walmart stores where inflatable floats with a dive flag can be purchased for like $15.
If you have a euro gun you will generally have better success because the fish here are skittish. But Big Island and all the islands other than Oahu actually have a reputation for having more fish that are a little less skittish, which has to do with the fact that Oahu is over-fished. I probably don't need to say this, but please be conservative and take good shots and only shoot fish you intend to eat. Depends on where you are diving and the food sources in that area, but you will probably see parrot fish (uhu), goats (kumu), smaller trevally (omilu), big eye emperor (mu), gray snapper (uku) and of course many other tropical fish but you may want to check first because some of these are prone to cigutera toxin or just may not be to your liking. Hope this helps.
I would have made the trip over there to dive with you, but I will be on the mainland that week. Check out the Hawaii Skin Diver forum. Sometimes guys from out of town post there and I'm sure someone would be up for a dive, maybe even lend you some gear.