Go to Settings>Apps & Notifications, then "See all [n] apps" to see your app list. Choose an app, then Notifications, then toggle the notifications on or off.
If you want a notification icon in the status bar but want to turn off the tone, then tap the word for the desired notification in that Notifications menu (not the toggle switch), then tap Silent.
If you want a notification icon in the status bar but want to change the tone, then tap the word (not the toggle switch), then tap Advanced, then Sound, and change the tone there.