1. Why it's worth making a complete backup
There are three moments when a backup makes the difference between a minor setback and a serious headache. The first is before installing a major operating system update: although these updates rarely cause problems, in some cases they can be interrupted or cause faults that force you to restore the phone to factory settings. The second is when you're about to switch phones, whether because yours broke or because you've bought a new one: the backup is what lets you have your photos, contacts and apps ready on the new device without having to set everything up from scratch. And the third, the most important one, is simply as a precaution: a phone can get lost, stolen, or broken by a knock or a drop in water, and in those cases the only way not to lose years of photos and data is to have a recent backup somewhere else.
The good news is that on current phones this backup can be fully automated. Once set up correctly, it keeps updating itself in the background without you having to remember to do it every week.
2. How to do it on Android
On Android, the backup relies on your Google account and is split into two parts that are worth turning on separately. On one hand, photos and videos are saved to Google Photos: open the app, go to your profile picture and check that "Backup" is turned on, choosing the quality you prefer (the "Storage saver" quality option compresses images slightly, while "Original quality" saves them as they are but takes up more space in your account).
On the other hand, the rest of the phone's data (contacts, calendar, call history, system settings, saved wifi networks and data from compatible apps) is managed from Settings → System → Backup (on some phones it appears as Settings → Google → Backup). There you'll see a switch to turn on automatic backup and, usually, the date of the last backup made. It's a good idea to always leave it turned on and check that the phone occasionally connects to wifi and a charger, which are the two conditions it usually requires to back up data overnight.
3. How to do it on iPhone
On iPhone you have two options, and they're not mutually exclusive. The most convenient is the iCloud backup: go to Settings → [your name] → iCloud → iCloud Backup and turn on the option. From then on, the iPhone will make automatic backups whenever it's charging, connected to wifi and with the screen locked, usually at night. You can also tap "Back Up Now" to force one right away, which is useful just before a major update or before switching phones.
The other option is a local backup with a computer, using Finder on a Mac or iTunes on Windows. Connect the iPhone with a cable, select the device and choose "Back up this iPhone now". This option has an important advantage: it doesn't depend on the space available in your iCloud account (which by default is only 5 GB free) and, if you check the box to encrypt the backup, it also includes saved passwords and health data, which the iCloud backup doesn't always cover in the same way.
4. What's included and what usually isn't
A complete backup, on both Android and iPhone, usually includes: photos and videos, contacts, call history, SMS messages, calendar, general system settings, saved wifi networks, the list of installed apps (so you can download them again) and, in many cases, the internal data of apps that support it.
What's normally left out, or needs to be checked separately, is different. WhatsApp, for example, has its own backup system, independent of the operating system, so it's worth going into its settings and checking that its backup is also on and recent. Passwords saved in the browser also aren't always backed up unless you turn that option on explicitly, nor is data from some banking or authentication apps for security reasons, which usually require logging in manually again on the new phone.
5. Cloud backup versus local backup on a computer
A cloud backup (Google or iCloud) has the advantage of being automatic, not depending on having the phone near a computer, and being available from anywhere if you need to restore it urgently. Its limit is the free space, which tends to run short if you have a lot of photos, forcing you to buy more storage for a small monthly fee.
A local backup on a computer, on the other hand, doesn't take up cloud space or have an added cost, and tends to be more complete when it comes to sensitive data like saved passwords. In exchange, it requires remembering to do it manually every so often and having the computer and cable at hand. The safest approach, if you don't mind spending a bit of time on it now and then, is to combine both: automatic cloud backup for day-to-day use, and an extra local backup before delicate moments like a major update or switching phones.
6. How to check that the backup has finished
On Android, go to Settings → System → Backup and look at the date shown next to "Last backup": it should be from the last 24-48 hours if you have the option turned on. For photos, open Google Photos, go to your profile and check that there's nothing left pending to upload.
On iPhone, go to Settings → [your name] → iCloud → iCloud Backup and you'll see the date and time of the last successful backup. If you've made a local backup with your computer, Finder or iTunes also show the date of the last backup on the device's info page. On either system, if the date is very old, check that wifi, charging and available storage space are all in order, since those are the three most common reasons a backup gets stuck halfway.
7. How to restore it on a new phone
On Android, when you turn on a new phone for the first time, the setup assistant itself will ask if you want to restore a backup and will ask you to sign in with the same Google account you used before. Choose the most recent backup from the list and wait for the data to download and, shortly after, the apps and photos (this last part can take a while depending on your connection).
Something similar happens on iPhone: during initial setup, the option "Restore from iCloud Backup" or "Restore from Mac or PC" appears if you prefer to use the local backup. Sign in with your Apple ID, choose the corresponding backup and leave the iPhone connected to wifi and power while the process completes, since it can take anywhere from a few minutes to a couple of hours depending on how many photos and apps you have.