Hanama Bay and Big Kahuna's are on Oahu. Many people have already listed the major things to do. If you have some extra time to kill I would reccomend walking around up in Waikaloa and walking around the Hilton, it is a rather amazing hotel. I think I will go up there one of these nights, its fun to wander around.
Going up to the observatories, or at least the visitor center, on Mauna Kea, is very interesting as well, but it does take awhile (about 2 hours one way from Kona) and many car companies don't let you drive on saddle road, which is quite bad.
Going up north is interesting, to Hawi or the various canyons.
The volcano is a worth while trip, take at least a full day for that, besides the long drive there is plenty to do and see there. Be sure to walk through the lava tube and if you have time drive down chain of craters to the ocean at night, when the lava is flowing it is an awesome sight.
There are not too many places to get surfing lessons here, but if you are interested your best bet would be Kahalu on Ali'i drive down by Keahou. There are a few surf lesson places down there with those big colorful boards.
As far as food goes, there are some good plate lunch places in Hilo but not too many in Kona. The only one that I can really think that is worth going to is down by Sack 'n Save, the name escapes me, I think it is Big Island something (I just got home, and haven't been down there yet).
Best diving is hard to say. I would be partial to Jack's Diving Locker, down by Hard Rock, but that is because I am friends with the owner's son. The night manta dive in Keahou is pretty amazing and the mantas are coming back as well. About a week ago there were 15 there, so I would suggest doing that. The other dive spots are all at the whims of the ocean.
I would skip the whale watching type of boats. There aren't any whales here and all you are likely to see would be dolphins, which you would probably see on a dive boat as well.
Big Island Forest and Trail have some interesting hikes if you are into that. You can probably hike on your own if you get a guide book or ask around. Most aren't too difficult to get to, you might just have to walk a bit further.
If you have any questions lemme know, I am here in Kona for another 2 weeks or so.
*Just a few ideas, I am one of those white locals, as it was put earlier, so might want to take this post with a grain of salt
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