Updated April 21st, 2020.
NOTE: the following applies to the iPad as well.
UPDATE: they’re up to 9.3.5 now.
Easy answer: if you’re already on some version of iOS 9,then yes, you should update. See which version your iPhone’s on by going to Settings, then General, then About. Look for “Version.”
If you’re on 8.x I recommend staying there unless you have a great reason to move up. Here’s Apple’s page telling you what’s new in iOS 9 compared to iOS 8. It’s not such an impressive list, and remember that moving to 9 from 8 will probably make your iPhone run slower. I wouldn’t risk it if I were you.
So, if you’re already on 9, the 9.3.3 update is for you. It’s full of security fixes: Apple provides a list of 26 security fixes in iOS 9.3.3, including issues involving the Calendar, FaceTime, and Safari. You need to plug these security holes. So get the update.
Always be sure to have a fresh backup of your iPhone before performing an update. Go to Settings/iCloud/Backup and check the time of your last backup. You can always do “Back Up Now” just to be safe. Here’s how mine looks:
Now that you have your backup, go get the update. I recommend you install the 9.3.3 update “over the air” which means you go to Settings, then General, and then Software Update. The other way (connecting your iPhone to your computer with the USB cable) is not only unnecessary, but slower too, so just stop it. Plug your iPhone into its charger, connect to WiFi, and download and install the update through the air. It won’t take long– a few minutes, a restart of the phone, and you’re done.
Just be sure you’ve backed up first.
Wondering whether to update your Mac? See this article about updating to 10.11.6.
If you have other questions, contact me. Use the form on my Contact page.