- What is 'safe' to drink? What makes water quality good vs bad?
The WHO has a 595-page PDF that probably contains more than you ever wanted to know about it.
Also note that water that's "safe" to drink for a person accustomed to it may be unsafe to start drinking suddenly. There's definitely some survivorship bias involved, but plenty of areas around the world have water that makes tourists sick but is tolerated by the locals.
- Are taste and safety/quality correlated?
Correlated, sure. Most bad-tasting water will be bad for you; most safe water will taste ok.
Bad taste, bad safety: stagnant swamp, sewage, seawater.
Good taste, bad safety: well water contaminated with trace amounts of lead or coliform bacteria
Bad taste, good safety: over-steeped tea, boiled water cooled to room temperature
- Any recommended water quality tests?
If you want a test, you can buy them online. Testing municipal water might be useful to tell you about what's getting added by the plumbing in your home or could help you dial in how much you trust your municipality's water quality reporting.
When I've lived in areas with municipal water, there have occasionally been boil water advisories because the water utility detected bacterial contamination in the water supply. This type of prompt detection and temporary mitigation requires ongoing testing and communication rather than a one-time test kit.
- To filter or not to filter?
Depends on what you expect will be in your water and how much you care about drinking it. If your municipality's safety guidelines allow a higher level of some contaminant than you're comfortable with, and you detect that it's actually present in your water, you'd probably want to attempt to remove it.
One thing to mention if you're going More Dakka on optimizing your drinking water purity is that there exist tabletop distillation machines designed for home use, although distillation is an extremely energy-intensive process.
- Which filtering methods actually work?
The effectiveness of a filter depends on what you're trying to remove from the water, the quality of the filter, the condition the filter is in, and whether you use it correctly.
One way to think of filtration is that the rural well water you were drinking before was distilled (evaporated and then fell as rain) and run through an extremely large biofilter (the soil) before it was stored in the underground aquifer that the well tapped into. This demonstrates that there exists an extreme of filtration which gives the water the traits you prefer drinking. The question then becomes "which filtering methods are feasible and yield worthwhile improvements to water quality at the home scale", which depends on your personal definitions of "feasible" and "worthwhile".
- What questions would you be asking?
I'd be asking what contaminants my municipality tests its water for and what levels they consider acceptable.
I would consider phoning the water provider and asking them very nicely "hey, I'm sure you get this a lot, but I spent some time on a farm lately drinking well water and now that I've come back I notice that the tap water smells/tastes pretty distinctive here. Could you help me understand why that is?".