There is a book " standard methods for the examination of water and waste water" that goes over most testing.
The best suggestion is to talk to the local waste water plant or water plant operater. They are usually friendly and can explain the tests.
The variety of tests depend on the problems in your area. ph and turbidity, definitely. Biological oxygen demand (bod) and chemical oxygen demand ( cod ) are two other common ones. If oil is a problem, you might also do oil and grease. The plant I worked at had a problem with zinc. Now the city had an electroplating plant that wasn't very responsible. So the city routinely did a bunch of metal tests. But there are many organic substances that can tested for also.
Testing protocol often goes like this. The samples that are pulled can be tested for something like ph. Then they have to be preserved ( usually acidified), refrigerated, and then sent to a certified lab for testing.