This first element is only infused at the White Sword later on the game.
Link gets the Fire Element after beating its guardian, Gleerok, in the Cave of Flames. This element, together with the Earth Element, when infused in the White Sword, will increase the strength of his sword.
Link gets the Water Element early in Temple of Droplets, by melting the ice surrounding it. But when Link melts it, it will unfreeze Big Octorok, the boss of the dungeon, who will steal the element. This forces Link to beat him in order to get it back. When infused, the White Sword will get a power upgrade.