I'm having trouble updating my phone even when my iPhone is connected to Internet.

Try following the steps here:

Update the iOS software on your iPhone, iPad, or iPod touch - Apple Support

If you are having issues with Wi-Fi try these:

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

Hi

Follow the instructions here, depending upon which method you are using to update iOS:

• Get help with over-the-air iOS updates - Apple Support
• If you see an error when you update or restore your iPhone, iPad, or iPod - Apple Support

If you are updating over-the-air, you may wish to consider updating via iTunes instead:

Update the iOS software on your iPhone, iPad, or iPod touch - Apple Support