If you never recorded one and want to record or change a voicemail greeting, you've come to the right place. The process is very simple. You'll just have to click the green phone icon, the same place you make calls, and navigate to the voicemail screen. On the voicemail screen, you'll see a greetings option on the top left.
iPhoneAlley details a technique to trick the iPhone into using any custom audio file as your voicemail greeting.

How to record or change your Android voicemail greeting in 10 simple steps 1. Turn on your phone and launch the Phone app. Turn the power on for your phone. Then, tap the Phone app. 2. Open the dial pad. Tap the dial pad icon near the bottom of the screen to bring up your phone's dial pad. 3. Call your voicemail.
When you reset network settings, all of your iPhone’s Wi-Fi and cellular network settings will be erased. This includes your Virtual Private Network (VPN) settings, your Bluetooth devices, and your Wi-Fi networks — so make sure you write down your passwords first! We do this step because it can be very difficult to track down the exact source of a software problem, so we reset all network settings.