Unlike other Tools, the Bow is not given to you at the start of the game. You acquire it by attending the first Archery Contest on Spring 28, Y1. It is given as a participation prize, and you recieve the recipe for Arrows there as well.
To use your Bow, press your "Use Tools" button (default Z on keyboard, X on X-box gamepad). You will a yellow line extend from your bow until it reaches max distance. The time this takes is your charge time. You can move while charging your shot.
You need Arrows equipped in your quickslots to use your bow. It is its ammunition.