You can go underwater in this game, provided you know what you're doing.
It is best to dive right next to the islands, as it is shallower there.
Typically, if you're going underwater, you want to drop an anchor, throw some shark bait, and jump in. Grab all of the clams, seaweed, stone, sand and clay you can and swim back to the boat, as you can only distract the shark for a limited amount of time.
You can get things which will improve your underwater experience, like fins and oxygen tanks, which are researchable items.