What various
forms of energy are present as you shoot a bow
andarrow?

In order to shoot an arrow with a bow, you should hold the bow with your left hand, and should pull the string to yourself and stay for a moment to aim.

At that point your arm applies a force, and that force does work on the bow. This work is stored as elastic potential energy. When you release the arrow, this potential energy is converted to kinetic energy, in this way the arrow is thrown forward.

This is the simplified version. Of course, in real life the case is different.

In real life, when you hold the string to aim, there is elastic potential energy, and also gravitational potential energy to hold the arrow above the ground. When you release the arrow, it goes through the side of the bow with a friction. So, part of the potential energy is converted to heat when the arrow and bow is still in contact. After they lose contact, the potential energy is converted to kinetic energy, and the motion of the arrow is projectile, since the arrow  both moves forward and downwards. Eventually the arrow hits the ground.