The Legend Of Zelda: Link's Awakening - How To Swim

It may not seem like it, but The Legend of Zelda: Link's Awakening is actually a fairly linear game. If you can't get somewhere, it's probably because you're not supposed to go there yet. Finish a dungeon or two and come back later. Eventually, you'll have the right tool for the job.

In other words: if you're exploring Koholint Island and your path is blocked by deep water or you see a chest stranded on an island that you can't reach, don't panic. It's not your fault. You simply haven't earned the flippers yet. You'll get the flippers, which allow Link to travel through dark-blue deep water, in Link's Awakening's fourth dungeon, Angler's Tunnel, as part of your regular playthrough.

Once you have the flippers, you're pretty much all set. Unlike the Game Boy version of Link's Awakening, Link equips his flippers automatically whenever he needs them. Just hop into the water. The game will take care of the rest.

While you're swimming, you can hit "A" to get a short speed boost, and "B" to dive underwater. The latter tends to be more useful. If you look carefully, you'll notice a few heart pieces submerged in Koholint's lakes and rivers (another change from the original game, which didn't show you what was under the surface, forcing you to rely on dumb luck in order to find the collectibles). Just swim over them and dive to pick them up. 

You'll also use "B" to swim under the barrier blocking the entrance to Link's Awakening's fifth dungeon, Catfish's Maw. Look for the bubbles. That's where you want to dive.