You should have a most recent backup of your iPhone before you update, in any case. Choose the method you prefer, iCloud or iTunes.
To update your device, connect it to iTunes. It's recommended in your case, as you're still running iOS 7x. You also won't need extra space on your device, because iTunes will erase everything before the update and install a fresh iOS version. If the 850 MB are not enough in spite of using iTunes, you'll have to remove apps you don't need any more.
Here are the instructions for updating through iTunes: Update the iOS software on your iPhone iPad and iPod touch - Apple Support
(Ignore the wireless version, you shouldn't update over the air.)