There are a few wrecks out on the south side of the island - the Waikiki/Ala Moana area. I've never been to them, but i believe there are two pretty close together that can be seen on one dive.
One is a fighter plane that the pilot was able to eject out of - apparently lots of people like to get in the cockpit for a photo op.
The Mahi is out on the west side of the island, in Waianae. It's about a 1 - 1 1/2 hour drive out to the harbor from Waikiki, but many operators provide shuttle service from Waikiki. Just dove it last week - it was nice - the group of rays that circle the wreck were the best part. Haven't seen any sharks. However, on the way to the second dive site (Makaha Caverns) we did see 3 whales hanging around - only about 20 feet away. We stopped to get a better look - way cool.
I did my OW certification dives at Electric Beach - it is a very popular site for beginning divers, and the parking lot can get very crowded. Best to get there early for a space. Depending on the time of year you come will determine how many shore dive sites are available. Many are on the north shore, which are really only diveable from April to November, depending on surf conditions.
Check out Captain Bruce's, who I've heard is a good dive operator. Haven't had a chance to go out with them yet, but they do dives out of the west, east and south harbors (something that other operators may not do). Additionally, based on the newsletters they send me, they do drift dives, "really deep" dives (don't know how deep that is) and other type of dives that the other operators may not do (because they're on a set schedule). Prices that I've seen average about $95 for a two tank boat dive, which includes tanks and weights.