No way to shut just the camera shutter sound off, but it will go silent with the other system/notification when you use Silent Mode.
You can also turn off the shutter sound in the camera app this way, or by turning on Live Photo. The second does not seem to affect screen capture, probably because you can’t make a Live Photo screen capture.
If you don’t see the bell icon in your Control Center, you can add it in Settings > Control Center.
Silent Mode does not affect music, and should not affect game and/or most app audio. Alerts, notification, and system sounds like key clicks will be silenced.