How can I know when new iOS updates are released?

ipsw.me provides an RSS feed of iOS releases at ipsw.me/timeline.rss. You can subscribe to this RSS feed (or a filtered version) using IFTTT. Templates are provided at ipsw.me/notifications. This is accurate to a couple of minutes.

You can also subscribe to the TSS status of iOS releases using the API available at api.ineal.me. This can be near instantaneous, providing a real-time view of TSS signing (and subsequently the releases).

By default you will already automatically get a notification when an iOS update is available.

Typically, after an iOS update is released, users should get a badge notification with a '1' sometime in the first 48-72hrs. However, an actual push notification may not come until quite some time after and only if your device hasn't been updated yet.

Using myself as an example, I have an iPhone 6s which I've had since September 2015 (it's now March 2017). In that whole time I've never had a push notification about an iOS update because I always update my iPhone fairly quickly. However, my wife who also got an iPhone 6s at the same time as I did, did get a push notification for iOS 10 at some point (weeks after iOS 10 launched) because she was still on iOS 9.3.5 and was in no hurry to update (I'm usually her guinea pig with these things). However, she did eventually update from iOS 9.3.5 to iOS 10 because the notification was relentlessly annoying her (and because iOS 10 was working well for me).

In summary

What I am saying is that a badge icon should appear within 72hrs of the official release, and this badge icon appears on the Settings app icon. Once this happens you can do an iTunes backup prior to actually installing the update. While updates may download automatically, they won't install without your input. In other words, you should never get caught out.