How do I get notification when a message is sent to me on my phone?

Settings > Notifications > Messages > Allow Notifications (and other options on that screen regarding the type of notification).

Do you have an Apple Watch paired with your phone? If you do the notifications will come to the watch rather than the phone.

You also won’t get notifications if the Mute switch on the left side of the phone above the volume buttons is toward the back of the phone, or if Do not Disturb is enabled.