While you can use the Wooden Sword throughout most of your adventure, there are two upgrades to your sword that you can grab to make them even more powerful.
Requirement: Min. 5 Heart Containers
From the start screen, head one screen right, five screens up, five screens right, up the stairs, two screens left and up the stairs on the screen with the waterfall. Enter the cave here to grab the sword.
2.0 Magical Sword
Requirement: Min. 12 Heart Containers
From the starting screen, head right one screen, up five screens, left two screens, down one screen, left one screen up one screen, left two screens, down one screen, left two screens, and up one screen to find yourself in part of the graveyard.
Push the middle grave in the third column from the left to reveal a stairwell leading to the sword.