Open Control Center on your Apple Watch
Tap the Silent Mode button
Turn on Silent Mode
To mute alerts only, go to Settings
Tap Sounds & Haptics
Turn off Alert Sounds
To silence notifications temporarily, enable Do Not Disturb from Control Center
To silence haptics, go to Settings
Tap Sounds & Haptics
Turn off Haptic Alerts
To mute the watch during a workout or meeting, use Theater Mode or Focus mode