Open the well cap. A simple hex nut holds most well caps in place, so all you need for this step is the properly sized socket wrench.
Pull the old submersible pump out of the well casing. With a shallow well, say around 50 to 100 feet or so, you may be able to pull it out by hand. Use a winch to pull the submersible pump out of the well for anything deeper than this
Detach the old submersible pump from the discharge line. This is a flexible heavy duty line attached to the top of the pump. This line runs through a pitless adapter casing below the frost line to the tank in your home.
Select a submersible pump that is proper for your conditions. For instance, a deep well may need a high powered pump to make sure the pump overcomes gravity. You also need to look at the GPM or the gallons per minute rating on the pump to be sure it has the needed flow rate.
Install the pump according to the manufacturer's specific directions. Be sure to follow frost line recommendations.
Dump a cup of bleach down the well to sanitize it and slowly lower the submersible pump down the well casing. Use a winch if necessary. Then replace the well cap.