Turks & Caicos for Reef Sharks. We dove with free-swimming, non-baited sharks on every dive at sites off French Cay. They were completely disinterested in us except to feed one night on the smaller fish attracted by the ships lights. I almost landed on one under the boat one dive. We did the Explorer but you can also get out there with a 45min ride from Provo. Here's a good idea of what to expect.
Saudio's Trip Reports - thanks to Mike Southard. It's also really excellent deep wall diving. T-shirt warm in the summer also.
I also agree with Doc, the farther out in the Bahamas you get the more sharks you see. In Cay Sal we always saw several per dive - one afternoon we quietly sat on the bottom (not a feed) while a dozen swam over/near us - As they became familiar they would get pretty close - passing within 3-4'. I don't know if anybody goes out to CaySal now - I think Spree did at one time. Maybe contact Wookie here and see what he's scheduled.
We've also done a Bahamas baited feed and the Cara/Cara dive off Roatan - there's a very different behavior exhibited when they know food is coming. Lots more circling and aggression. When they tell you not to wave your arms around, that's not because the operator knows what to expect. The Cove's feed info is here:
http://www.stuartcove.com/DiveBahamasMgmt.aspx?id=2&pageId=65
If you like that sort of thing one of the best is Unexso off Grand Bahama. Cristina currently works there:
http://www.youtube.com/watch?feature=player_embedded&v=WK2LpUoqX6A
JASA baits the Tigers/Lemons in also - watch the Youtube videos - you'll see Abernethy do it. I believe other boats do as well. Cove's now runs day trips out there from Grand Bahama - they do also - there's a picture showing it here:
http://www.stuartcove.com/tigerbeachMain.aspx?id=8
I couldn't even tell you where to start looking but there's some sort of Lemon Shark research done off Bimini. I watched Anderson Cooper dive there once with a bunch of them. It seemed more wild - and a
lot of sharks in very shallow water.
Best place to dive with Whale Sharks off North America is Gladden Spit off Placencia Belize. Only during the next several full moon cycles - the last one might be mid-june - it changes yearly. My buddies dove with 8 last year. Splash Dive has details on their website.
Pricier would be Cocos Island on Sea Hunter, the Galapagos on any number of boats or the Socorros off Mexico. The Socorros might be winter-spring diving only though - IDK if anybody goes out there in the summer. More sharks of all kinds than you'd think possible. Solmar V or Rocio Del Mar go out there. Watch the video at the bottom of the page:
http://solmarv.com/ I'd guess most of those trips are already full for this summer, they're pretty popular destinations. Maybe you could find a space if you're a single diver.