why do i get cold water during the shower every time the pump kicks on
I would check the piping to the new hot water tank to make sure that the installer did not switch the cold inlet and hot outlet. What you explain suggests that the installer moved the cold water inlet to the hot water side so that when the well pump kicks on the cold water is being delivered into the hot supply piping from  the tank and not into the cold inlet supply. If you could, post pictures of the tanks piping and label the pipes.