First, you'll need to enable sharing for apps that support the feature. Kelly Guimont explains how to do that for Safari and other apps, and I show you how to do it for Photos extensions.
Use the list handles to rearrange app order in the Sharing activate view
You can rearrange app icons from the same view where you enable or disable specific apps; just tap and drag the grippy-looking three-bar handle at the far right of the list to move your favorites higher.
Tap and swipe in the Share Pane to rearrange icons, too
To change the order from the Share Pane, just tap and hold an app's icon, then slide it left or right to place it right where you want.
I moved the shared app icons I use most often to the left so I don't have to swipe in the Share Pane to find them. It saves me time and cuts down on the number of gestures I need to move content between apps.