The big three costs
Most of your monthly pool spend comes down to electricity (pump + heater), chemicals, and water/evaporation. Everything else is small by comparison.
Electricity is usually #1
The pump runs every day, so it's often the single biggest cost. Switching to a variable-speed pump can cut that dramatically because it runs slow and efficient most of the day. If you heat the pool, a heat pump is far cheaper to run than gas.
Chemicals
Chlorine or salt, plus pH and alkalinity balancers, make up your chemical budget. A cover reduces chemical loss from sun and evaporation, so you use less over the season.
How to lower the bill
Three moves save the most: a variable-speed pump, a cover to slash evaporation and heat loss, and a robotic cleaner so you filter efficiently instead of running the pump longer. Each pays for itself over time.