You can start paying immediately with your phone, even if you don't have the plastic card in your hands. All you have to do is activate the card in Google Pay or Apple Pay.
Instructions for activating Google Pay.
Instructions for activating Apple Pay.
You will see your card number in the card screen when clicking "View card number". If you then click on the card number, it will be copied and therefore be very easy to stick in the right place when you are shopping online.