iPhone X visual/taptic notifications

Just got the iPhone X and I noticed that when I receive an iMessage/text message, my Apple Watch vibrates but my phone doesn't display anything and doesn't vibrate to signify there is a new notification/alert while the device is Sleeping.

This happens with Mail, the Gmail app, etc. I get a new email, the Watch vibrates but the phone remains dark until I wake it up. Once it is awake, I can see the list of messages/emails and while I am interacting with it, if anything new comes in, I will get the notification at the top of the screen and it will vibrate.

Coming from a 6 plus, i'm sure there is a learning curve but if I didn't have the watch, how would I know if I received any messages/email? Do I need to constantly wake the phone up to see if I have anything new?

I have gone into Settings and made sure 'Do Not Disturb' is turned off.

Any help to get the phone to wake up and show/vibrate new notifications when they come in would be much appreciated.

Thanks!