Phone damage

Depends on the damage, but it is not going to be easy to extract data direct from the phone if you can't use the touchscreen, because any method will require you to unlock the iPhone by entering your passcode.

If you are lucky you already have most of your data backed up or synchronised to iCloud. If this is the case simply go through the new phone setup, use your existing Apple ID then accept when you are asked if you want to use the iCloud backup. I have done this when I replaced my drowned iPhone 6s Max, it worked perfectly and even offered to migrate my Apple Watch to the new phone. The only thing I had to do manually was re-authorise my cards for Apple Pay.

(Hint - if your iPhone is the trusted device for SMS security codes don't setup Apple Pay during the initial setup, skip the step and wait until after the phone is activated. Otherwise during initial setup the code is sent to a phone which is not yet able to receive, and it will time out before you can use it.)