You need two float switches in the poly tank.
There will be a Normally Closed switch (closed when hanging down) near the top of the tank that will control the well pump.
There will be a Normally Open switch (open when hanging down) near the bottom of the tank that will protect the pressure-tank pump if the poly tank runs low on water.
I usually have the switches operate definite purpose contactors (big relays) to run the motors. I mount the contactors in a small pull box, such as an 8"x8"x4" box.
I get the switches from USA Bluebook in Chicago, and the contactors from Grainger.
You have several choices for the second pump. You can put a submersible in the tank, or use a multistage centrifugal booster pump, or a shallow well jet pump. The submersible is most efficient, the booster next, and the jet the least efficient. Power cost is usually not an big factor in that application.
You should decide if the existing 10 GPM is adequate, and determine at what pressure you want to operate the pressure tank. Then select the pump.
If you ever need to replace the well pump you can select one with a little less pressure capability because you are not pumping to a pressure tank.
There is always a risk of contamination in a non-pressurized tank. You will want to watch for algae growth and add a little chlorine (laundry bleach) if necessary to keep it controlled.