In preparing for upcoming international travel, I have read advice to make a copy of my passport and not carry the original when walking around town. Is a copy of the first page enough, or the first page and the entry page, or is a copy of every page needed?
Having a copy of your passport is a good idea. I've been doing this for decades and it was useful on multiple occasions. By default, I copy the two main pages facing each other. The most important is that it shows the passport number, full name, photo and expiry. My advice is to make more than one copy and store them in two different places, in addition to having the passport stored in a safe place.
There are many countries that require a visa based on my nationality. When I am traveling to any of those, I also copy the page where the visa was printed on. Perhaps for some strict copies you would also copy the entry stamp page but that would be a hassle since you would have to find a place to copy the after you enter the country.
One more copy that is practical to have is a digital copy that is accessible to you online. If for some reason you lose your belongings, you can go to an internet cafe or business center and access you digital copy to print. That way you can almost always manage to have a copy should you need to show it to an embassy or police station.
For backup purposes, all you need is the photo ID page with your passport number, legal name, expiry date etc. If you lose your passport, a copy of this page is sufficient for making police reports, applying for new passports, etc.
Some people (and, in some countries where required by law, hotels) also copy visas and/or entry stamps for the country to provide additional proof they're in the country legally, but I personally have never bothered to do this: Immigration can easily look up your status as long as they have the details on your passport ID page. YMMV.
I've never needed to copy an entire passport, and about the only use case for doing so would be if you need to prove to somebody afterwards (tax authorities, unusually persnickety visa applications, etc) that you've spent X amount of time outside the country or visited Y countries on Z dates.