STOP: Before doing anything that might cause data loss, make a backup of your iOS device to either iCloud, your computer, or both for extra security—>How to back up your iPhone, iPad, and iPod touch - Apple Support

Check compatibility:

Go to Settings/General/Software Update, and it will tell you the most recent version of iOS that your iPhone is eligible for. In some cases immediately after a new version is released you will be given a choice of an update to the current version or to the newer version; for a brief period you can choose either.

If your phone is compatible, for most users, just go to Settings/General/Software Update or follow Apple's update support article→Update your iPhone, iPad, or iPod touch - Apple Support

If you have a problem updating with your computer (rather than over the air) the first place to go is→ If you see an error when you update or restore your iPhone, iPad, or iPod - Apple Support

For over-the-air updates, here are the conditions that must be met to update iOS:

You must have a fast Wi-Fi connection; you cannot update over cellular data unless you have a phone that supports 5G (series 12, 13, 14, 15, 16, 17 and SE 3rd edition) and you have a 5G connection

You cannot update using an iPhone personal hotspot

You must have a cellular network connection with cellular data enabled so the account can be verified after downloading the update

The phone can not be in Low Power mode (how to tell: The battery icon will be yellow in low power mode)

The battery state of charge must be above 50%

You must have at least 6 GB of free storage (for iOS 14 or later), and more is better

You cannot be connected through VPN

If you are not planning to update automatically overnight you should turn this option off, as it will only update when the phone is plugged in, connected to Wi-Fi and locked

If you do want to update overnight your phone must be connected to power, Wi-Fi and it must be powered on and locked overnight - if it doesn't update go back to doing it manually so you can see any error messages

Even for manual updates you should leave the phone plugged in until it finishes updating, both to assure that it remains above 50% and to keep Wi-Fi enabled (Wi-Fi turns off when the phone sleeps if not connected to power)

Here are a few special conditions:

Finally, if nothing you do works for an over the air update you can always update using your Windows PC or Mac. It's described at the top of the page in→Update your iPhone, iPad, or iPod touch - Apple Support

Here’s more info from Apple→If your iPhone, iPad or iPod touch won’t update - Apple Support

REALLY Finally, if you have given up hope either post as many details as you can think of in response to this post, or contact Apple's support team using the Get Support link at the top of every page.