To be able to dive, you should obtain a Wet Suit first. This can be purchased from the Tropical Island for 40 medals. You can earn medals by completing various tours / mini-games. (More details on the Tropical Island section of the guide) You can also use the free Rental Wet Suit by talking to Lloid outside the bungalow. Since it is a rental suit, you can only use it while you’re on the island.
First, wear the Wet Suit then head to the beach. Press A when you’re near the water to swim. To swim, you have to keep pressing A until you find a black spot on the ocean floor. Bubbles will also give away the position of some sea creatures you can grab.
To dive, press X. You must “touch” the black spot to get it. Take note that some spots move away and different speeds. This is true for animals such as the octopi and crabs. You can move underwater by pressing A but not as fast as you are in the surface. If you want to catch up, I suggest swimming in the surface first then when you’re past the target, dive and nab it.
There’s only a limited area around the island so if the shadow reaches the barrier, you won’t be able to catch it.
I have never seen the orange or white wetsuit on islands and I've been on them about 20 times a day. Wierd.