Mantas are very much a Big Island thing (I hear you can get lucky in Maui, too). As for turtles, it doesn't seem to matter which one of the islands you visit - turtles seem to be a staple of diving here. Your chances of seeing a Green Turtle during a dive here on Oahu are excellent. Seeing multiple turtles during a dive is fairly common, in fact!
I can't speak for the other islands, but there are dive sites here on Oahu where white-tip reef sharks like to hang out. My sightings have been confined to the south and leeward sides of the islands: Airplane Canyon (leeward), for example, has a little dude under one of the ledges. There are also little dudes at the Baby Barge (south side). I've never seen a Tiger shark, though, but I don't think that's the kind of shark you wanted to see, is it?!
If you're really lucky, you may get to see humpback whales, since you are coming to Hawaii smack in the middle of the humpback whale migration season. The critters migrate from the North Pacific to the main Hawaiian islands during the winter months, usually starting around November. I've been fortunate enough to see them from a dive boat, but I haven't been lucky enough to hear them singing underwater, or better yet, catch a glimpse of one while I'm in the water. Those that have say it's an amazing experience. There are great vantage points on Oahu, Maui and the Big Island where you have a chance of spotting them - see
Hawaiian Islands Humpback Whale - Marine Life - Top 10 Whale Watching Sites There are dive sites on Oahu's south shore that are close to the vantage points mentioned by NOAA, so you might get lucky if you hit those spots.