One of the coolest things about your iPhone is that, unlike other smartphones, installing a new operating system is extremely easy. These updates keep coming regularly to add things such as small improvements, performance and security fixes among many other things. When it's a major release, the new software will even give you the impression you're using a new device. Every new version of the iOS operating system has been tested already, so it's always recommended that you install the latest build when it becomes available. But how do you do it?
Whenever a new update is available, you'll get a message asking to you update. Since this can be dismissed fairly easily, you should look for a red badge with a number one on the Settings app. That's going to be the only time you will see a badge here, so it's a very clear indicator. Simply tap on the Settings app > General > Software Update. All the buttons should show a number one to indicate a new update is available.
Download and install directly from your device
If you miss all these steps you'll be prompted again the next time you plug your iPhone or iPad to your computer. I've found a lot of friends with iPhone prefer to use their device independently and never get to plug it unless they want to backup some pictures or transfer some music. To update your operating system directly from the device, simply check you have access to Wi-Fi network to download the update and possibly plug the device to the mains if you don't have much battery. You might need to make some room for the new update to install as well, as it will need some storage available.
Once you tap the "Install Now" button you'll have to accept the terms and conditions and wait for the update to "verify" this is a legit device. Once all those messages are accepted, the actual download and installation will begin. This will take a while. Once the update is installing you will see the black screen with an Apple logo that you normally see when you restart the device. Disregard the painfully slow progress bar, which means you need to wait some more time without being able to use your phone. Tip: you can do this before going to sleep so you don't miss any important call.
Since you're going to be connected to a Wi-Fi network, it might be a good idea to back up your device to iCloud before the software update. You can also do this locally on a computer using iTunes, but that defeats the point of a post-PC device, right?