Your Cargo Lift will automatically travel up and down the shaft if given enough power, and will collect and offload resources depending on the orders given. To change the orders, simply open the Cargo Lift Controller UI (default key is 'E') and select them appropriately.
Once at the upper or lower bounds of your shaft, the Cargo Lift will attempt to Load and Unload via any attached Storage Hoppers. Simply place them touching the parked lift.