A bunch of scientists around the US interested in stream biogeochemistry settled on this list to monitor around the country:
fraction dissolved organic matter
From (see): http://pulseofstreams.weebly.com/products.html
The US EPA has a manual; it's not super up-to-date: https://www.epa.gov/sites/production/files/2015-06/documents/stream.pdf
The USGS also has a report: https://pubs.usgs.gov/fs/fs-077-02/
Yes, a fair bit of what matters and what you would expect to find in a healthy stream varies a lot by location; local environmental agencies, or local branches of federal agencies, and local nonprofits (and academic researchers, for that matter) may be able to provide you with location-specific protocols and advice.
A perhaps useful generalization that I've learned from my own experience (I'm an aquatic ecologist, mostly) in terms of practically choosing what methods are available in any given situation is that ecological community data, based on identifying species, is an option for when you've got time and, ideally, relevant expertise (i.e. people who know how to identify bugs- here's a website meant to help with that: http://www.macroinvertebrates.org/, and there are some citizen science apps that help with identification that have gotten pretty amazing, like iNaturalist/Seek: https://www.inaturalist.org/, https://www.inaturalist.org/pages/seek_app- but bugs are just hard to identify, compared to, say, vascular plants, and usually require a microscope to identify with much precision), but perhaps no additional money, whereas chemical data usually requires at least a little (sometimes a lot of) money for the sensors or tests. (This dichotomy influenced what got done in my own dissertation, based on available resources- my own time, not so much money- quite a bit.) Some geomorphology data can also be collected very cheaply, like with just a normal ruler.
Any of those approaches (and others) can yield useful information; it's just different. I know stream biogeochemists (people who made the StreamPulse project, above) who insist that dissolved oxygen is the most important parameter, because it reflects all the life activities, all the respiration and photosynthesis (the latter requires collecting data overnight, or in dark bottles), occurring in a stream. I know ecologists who insists that bugs (usually benthic macroinvertebrates) are the best, because they reflect multiple trophic levels (i.e. a lot of the food chain), and many of them are sensitive to water quality and other parameters, so what is and isn't there indicates water quality etc. There are "Indices of Biological Integrity" (IBIs) specific to different locations that score stream health based on species presence and abundance; if you can find one for your area, it might help direct you as to what specific community data to collect, or at least what organisms matter. An emerging technology, which may eventually build on IBI-style community analysis, that isn't very accessible now, but may become the standard (due to high information to effort ratio, as analytical techniques progress) in decades to come, is eDNA. For eDNA, you collect a water sample, then isolate, amplify, and analyze all the loose DNA fragments in it (which for now requires a genetics lab and highly skilled bioinformaticians, but maybe someday the technology and code will be widely available and routine, kind of like how genome analysis has gone from an extremely expensive thing that took years and boatloads of money to something you can get for your dog for less than $100; someday both will probably come from sensors with instant feedback), and can potentially get an idea of all the organisms that have been in the stream (at least upstream) recently. Stream restoration tends to focus mostly on geomorphic parameters, e.g. meander bends and rocks, because those are easy for developers to control, so there's a community of people who care most about that kind of thing. People at water treatment plants or government environmental regulation enforcement agencies tend to focus on water parameters that have legal regulations around them (think Clean Water Act, endangered species), often relevant to human health. These kind of things tend to get a lot of monitoring already, especially in larger streams (small streams are less well-monitored in general), and have well-developed protocols and widely available equipment for their methods, because it's legally required. The laws, regulations, and even monitoring protocols tend to lag behind the science, because it takes time and political power to change them. Some industries/chemicals/issues, like pharmaceuticals and personal care products, are usually not regulated, not treated for, and not monitored, and so are very poorly understood; reasons may be political, but also procedural; testing for, say, illegal drugs in urban streams, is a much more specialized task at the moment than testing for nitrates. Social scientists interested in human well-being would probably have a whole other perspective- besides testing for medically relevant water quality (toxins, pathogens), I've heard of counting toothbrushes left by homeless individuals, for example. OK, I'm rambling; the basic point is that different people, including experts, will have different opinions as to what data matters most, depending on the intended use of the data.
Here's a series of stream-monitoring activities executed multiple years by high school students in Durham, N; they're fairly low-budget, not super time-intensive, and fairly easy for non-experts to do: mohttp://web.archive.org/web/20170523061110/http://www.learnnc.org/lp/editions/mudcreek/ (I helped supervise these for a few years, and they worked well. Friends of mine adapted them for use with middle-schoolers.)
WikiWatershed also has advice and tools: https://wikiwatershed.org/
You may be able to augment your own data collection with existing data, particularly if your local stream is large, people are drinking from it, etc. See the US Water Quality Portal to start: https://www.waterqualitydata.us/ (or the perhaps more user-friendly "How's My Waterway?" from the USEPA): https://watersgeo.epa.gov/mywaterway/mywaterway.html
I guess I should back way up and say, you might also want to consider what is feasible for you (financially, in terms of time requirements, in terms of how long or how many times you have to collect data before it means much, in terms of what expertise you have available), to help you narrow down what data to collect; I wrote the above with the assumption that feasibility matters. If you have unlimited resources, wow, I wouldn't even know where to start. Good luck!
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