Apple - Which folders should be copied when performing a manual backup?
The /Users folder is the most important, thats where all your personal data is kept.
The items in /Applications folder can always be re-downloaded but won't hurt to back them up too. Pay special attention to the apps that can't be easily obtained by downloading from the Mac App Store or from their web-site, like older version of app, an app no longer available from the developer/publisher or a self build app.
Like most things in life, I think it depends :-)
At one extreme, you might want to back up absolutely everything, so that in the case of a disk failure you can restore to precisely where you were: every app, file, setting, configuration, &c. This isn't really practical to do manually. (It'd need root access, knowing exactly which vm/cache files to exclude, what to sync first, &c.) I use Carbon Copy Cloner and SuperDuper! to create clones, and Time Machine for incremental backups.
At the other extreme, you might only want to back up the most irreplaceable files (e.g. documents and/or photos). This should be easiest, but will of course require the most work in the case of a failure (re-downloading and reinstalling apps, re-setting up all the config, &c). For this, you'll know where the files are; it's easy to drag them to another drive.
For something in the middle, it depends what you might have changed and how much effort you want to put in…
I think the question has already given the three most important areas: if you back up /Users, /Applications, and /Library, then you'll probably include all the important stuff for most people.
If there's anything vital elsewhere, then you'll probably know about it, because you'll have set it up yourself! Examples might include: programs you've compiled and installed in /usr/local (or got Homebrew &c to do for you); hosts file entries you've added to /etc/hosts; directories you've created under /; locales you created in /usr/local/share/locale; daily/weekly/monthly scripts you created or linked under /etc/periodic/; a message-of-the-day you wrote to /etc/motd; and so on. (If you've disabled System Integrity Protection, you could even write under /bin or /usr/bin…)
Also bear in mind that apps, drivers, and anything else that's not plain data will need care to preserve user, group, permissions, ACLs, &c; most external filesystems won't fully support all of those. And you may need root access to back up and/or restore some things. All in all, it's much easier to leave it to a dedicated program!
Finally, this is a good opportunity to remind everyone of the importance of backups. There's no single correct strategy; everyone's needs are different. But please think about what would happen if a disk died, and do as much or as little as you need to prevent that becoming a disaster.