Hitting a speed bag isn't very useful for punching speed or power. I would hit a heavy bag, a target bag, or focus mits.
It depends on the punch to some degree but basically you will use your thighs, your hips, your abdominals, obliques, pectorals, deltoids, triceps, and forearms. You will also use your back muscles and bicep muscles to some degree. In short, if you throw a proper punch you will use most of the muscles in your body. Follow a solid strength training program that hits each muscle group. Perform the eccentric contraction slowly and the concentric contraction as fast as safely possible. Maybe add a few plyometric exercises like Explosive push-ups. Hit the heavy bag, target bag, and/or focus mits if you have a partner. Start with 3 x 3 minute rounds and add a round each week. You're punches will improve a lot.