For example, if I take a slap shot on a hockey puck, from what I understand, the forces acting on the puck are friction, the normal force, and the puck's weight. And, since I'm not constantly either pushing the puck, or pulling the puck there are no other forces acting on the puck, because me shooting the puck was just a one time force. Am I thinking about this correctly?
Yes, you are thinking about it correctly. No force is required to keep the puck in motion. This is an important idea in physics. It is actually a common misconception among physics students that a force is required to keep an object in motion, so it is good you do not have this misconception.