So, I don't get on here very often anymore, but this seems like a topic that comes up fairly frequently. I thought I'd put together a post on what's worked for me since moving down here a couple of years ago, and maybe help some fellow fisher's out if they're traveling to the islands and want to wet a line while they're here.

Gear
A cheapie setup from wally world or sports authority will get you going just fine, something med. action 10-20lbs., 6-8' will cover the most bases. If you're bringing rods and reels from home, most steelhead and salmon rods will get the job done. I prefer 20-40lb. braid for everything except plunking, then I use mono, again in the 20-40lb. range. Probably not going to bring any monsters in on these setups, but they'll do for casual use.

Lures

I've hooked fish on all of these lures, each one has a particular set of conditions that it works the best however. Swimming plugs like the crystal minnow (red/white) and the x-rap work better in deeper, calmer water for me. A nice protected cove, jetty, or harbor are ideal for these. The retrieve should be fast and jerky, almost fast enough to make the plug spin out, with pauses here and there.

Top water plugs, the black/red eye on the bottom and black/silver skirted one on the left, work great in rough, rocky areas with a lot of wave action and white water. Find a rocky point to cast off of, and try to cast past the breaking waves and rip it into the whitewater. The retrieve should be really fast and jerky, the plug should be splashing and gurgling, leaving behind a bubble trail and generally looking like a panicked baitfish.

The top two lures are my go-to's for almost any spot. The brand name is scrounger, they are a jig head with a plastic lip that makes it act like a swimming plug. The hook rides upright, so they are good for shallow spots because they won't get snagged very often. The best all-around is a 1/2 oz. scrounger with a black 4 1/4" curly tail grub, cast into little whitewater pockets in shallow coves and points with a fast retrieve. Or on sandy beaches, slow the retrieve down and let the jig bump the bottom on occasion, kicking up sand.
Bubble float/egg weight setup.

This is probably the most popular setup for light gear fishing down here, what people call whipping. Run your mainline through the float, through a bead, and tie on a barrel swivel. You want as long of a leader as you can manage to cast, usually not longer than your rod. You can use either a standard j hook, around a size 2 or so, or an AH hook, which is like a cross between a j and circle hook. AH is what most people use, around size 14 or so. For the lure itself, you can use minnow strips, which are just thin strips of plastic, or curly tails, sized according to your hooks. Thin strips of squid also work very well, although they don't last very long. You can slide the center of the float out to fill it with water to give you enough weight to cast. Retrieve should be fast, making the float splash and kick up a fuss.

If you're in a sand area, replace the bubble float with a 1/2-1oz. egg weight and drag it in along the bottom. If you're in a calm, sandy bottomed spot (lagoon, harbor, etc.), try still fishing this setup like you would for planter trout in a lake, using squid for bait.
For kids, replace the sliding bubble float with a fixed float, a couple feet of leader with a split shot ~6" from the hook and bait with small strips of squid or day old bread, rolled into balls.
If you're in a rocky/reef area fishing at night, try using a smaller hook with 2" glow in the dark grubs.

Bait
For plunking, or what everyone here calls dunking, your standard three way swivel plunking setup will get the job done. You'll need either a sand or rock spike to hold your rod, depending on where you're fishing, both of which you can pick up most anywhere for around 10-20 bucks, and a bell to clip to your rod. Circle hooks are the way to go, not only do they set themselves while the rod is in the holder, but they are much less likely to get hung up on the coral/rocks then a j hook is. The most popular style is called a BKN, around a size 14 will do the best all around.

Your leader shouldn't be more than a foot long, if it's a rocky/coral spot, as short as 6" is good. This will keep your hooks out of the reef, and makes it harder for an eel to pull your line into it's hole and end up loosing the whole setup. The lead line should be short in sandy areas, usually about twice what your leader is, so that your bait is lying on the bottom. For rocky spots it should be as long as you can cast, at least 6+ feet. Bank sinkers between 2-6oz. should hold in most spots. The best, cheapest, and easiest bait to get is the boxed squid from the grocery store, cut into strips according to your hook size. Octopus, chunk bait like tuna belly, and shrimp all work very well also.

Spots
Just about every inch of shoreline will hold some type of fish. There are three basic types of shoreline, sandy beaches, coral lagoons, and rocky/lave shoreline and cliffs.

Sandy shores will hold bonefish, jacks, threadfin, and barracuda. Popular surfing and swimming spots are good areas to look for bonefish, if you can get there before the hordes. Whipping with an egg weight and squid strips works well in these areas, cast past the breakers and work back in to the beach. Dunking is also productive in these areas, esp. at night. In the spot above, I'd work a lure in and out of the whitewater in the red circle, and dunk bait or whip an egg sinker rig in the yellow.

There aren't any real lagoons in the state, but Oahu, Maui, and Kaui have fringing reefs that will block the surf and allow extensive inshore coral reefs to form that hold all kinds of fish. In these areas, look for sandy channels between the coral heads that open into deeper water past the reef. Predatory fish use these channels as a highway coming into and out of the reefs with the tides. Any and all of the tactics above will work here, just be aware that coral is incredibly sharp and will cut your line like a hot knife through butter. If you're fishing at night, expect plenty of eels.

Rocky/lava shores are more prominent on the younger islands, so Maui and especially the big island have plenty of these. This can range from shallow coves not more than a couple feet deep at high tide, to the cliffs on the east and south shore of the big island where you can cast into 300'+ of water from shore. You'll need specialized gear to fish off the cliffs, not to mention the safety factor, so look for spots that you can get within a couple of feet of the water. These spots are where the scrounger works the best, if it's deeper than 20' or so you'll have more luck dunking bait on the bottom.

Here, I'd cast a scrounger to the red spots, making sure to work it near the boulder that's circled in yellow

Cast to the spots on the edge of the whitewater and work back in. It doesn't hurt to leave the lure in the water for a few seconds when you get it back in, right at the edge of the rocks. Sometimes fish will follow your lure all the way back, and only attack when they think they've got it cornered against the rocks.

Times/Tides
Tide swings here are tiny in comparison to the NW, but they still have a tremendous effect on the bite. Just like jetty fishing, an hour or so on either side of a slack tide, especially high, is the best. If you can fish the first or last light of the day, do it regardless of what the tide is doing. After dark, I don't think the tide is as important, but still won't hurt to have fresh bait on if the tide is about to change. Expect plenty of eels at night anywhere there are rocks or coral, they are nasty buggers, so it's usually best to just cut the line near the hook and let them slither back into the water.

I've got some pictures I'll put up later of spots so you can get an idea of what to look for.
Hope that someone will find this useful, if you want to get more in-depth look for any of the Fishing Hawaii Style books by Jim Rizzuto, or Pacific Shorefishing by Mike Sakamoto, both full of good info.