Trying to update 15.3.1 and continue to get Internet error message, but I am connected to Wi-Fi.

Check first the range of the mobile network, see this Apple article: If you see No Service or Searching on your iPhone or iPad

If you see No Service or Searching on your iPhone or iPad - Apple Support

Check also the Wi-Fi network, see the following Apple articles:

Connect to Wi-Fi on your iPhone, iPad, or iPod touch

Connect to Wi-Fi on your iPhone, iPad, or iPod touch - Apple Support

If your iPhone, iPad, or iPod touch won’t connect to a Wi-Fi network

If your iPhone, iPad, or iPod touch won't connect to a Wi-Fi network - Apple Support

If they are On, turn Off Hotspot & VPN.

If you are home, reboot your router, WiFi station, and then your phone.

I’ve found rebooting devices often fixes most things. If you’re on a public network or reboots don’t work, hopefully someone else will have some suggestions.