Is there any cheap way to test the quality of my water? I did some Googling around, looks like there are services that will do this for a fee but didn't look particularly affordable.
If you are on well water, call your county department of health, there might be a fee, but it should be lower than a private lab.
I'd expect them to work reasonably well, at least for lead. The chemistry is pretty simple. Whether they can detect low concentrations is another matter. You could boil a pot of water until most of it is gone and test that. You'll have a much higher concentration of any contaminants.
An example: https://www.homedepot.com/p/Safe-Home-Do-it-Yourself-Lead-in...