The Watch isn’t an always on device and the screen only switches on when it thinks you’re using it. When it is on your wrist, it switches on when you lift your hand towards you and it quickly switches off when you put your hand down. It is pretty clever but occasionally it doesn’t switch on when you needed it to be. However you can always wake the screen manually by tapping on the glass.
By default, the Watch mirrors the notifications from your iPhone and it knows if you’ve read it so there’s no duplication on either side. When a message comes in for example, WhatsApp, it knows when you lift your wrist and it will automatically display the message without having you to press or tap anything. This is quite useful if you’re stealing a glance while your hands are full. This however won’t work if you have more than 1 unread WhatsApp messages pending, and you would need to view them individually from notifications.
For a hands-free way of calling Siri, you can do so by saying “Hey Siri” the moment you wake the screen. Some of the useful things you can do is to ask Siri to set an alarm or to remind you of your next appointment. You could also ask Siri to set a timer for 10 minutes or to create quick reminders on the fly.
For notifications that can’t be viewed on the Watch, they are pushed to the iPhone via Handoff. This would appear as an icon on the iPhone lock screen. To view it, just swipe from the left bottom corner similar to quick launching the camera.