Link won't be doing much, if any, under water exploration in The Legend of Zelda: Breath of the Wild. Diving isn't doable unless jumping from a high ledge, and even then his reach under water is very limited.?

The best way to get around water is by using the Cryonis Rune, or by using wooden rafts with sails in conjunction with a Korok Leaf. Equipping the Zora Armor slightly increases Link's swim speed and lets him swim up waterfalls to get around more easily, too, but your total Stamina will still limit the amount of time you can spend swimming. The more Stamina you have, the longer you can swim. If you don't have the Zora Armor, you can use the Cryonis Rune to freeze blocks of water on vertical surfaces to travel upward.

To dive, run towards ledge, press jump as you are reaching ledge and press action button, even though nothing will be shown on the screen.

Note that Link cannot stay underwater even after diving from up high. In order to reach objects below the water, you will have to either use the Magnesis Rune to bring metal objects out of the water, or place a Cryonis Rune under non-metal floating objects to raise them up.