leaking antifreeze all over the place?

Hello, I'm Danny.

The best way to test for leaks is to perform a cooling system pressure test. Without the car running it pressurizes the system with air and you can then see where leaks are coming from. Here is a tutorial showing how to perform a pressure test:

https://www.2carpros.com/articles/radiator-pressure-test

If the leak does turn out to be coming from the water pump here are a few tutorial showing how to drain and flush the coolant as well as how to replace the water pump:

https://www.2carpros.com/articles/coolant-flush-and-refill-all-cars

https://www.2carpros.com/articles/water-pump-replacement

I've attached picture steps below on how to replace the water pump on your S10. Hope this helps and thanks for using 2CarPros.