I need to transfer my contacts and messages from my old broken iPhone to my new iPhone. How do I do that?

You can do it as long your old broken iPhone is still functional. You can do a migration of data; an option available on iOS Setup Assistant screen during setup of your new iPhone. Or, make a backup to iCloud or to your computer of your old iPhone and then restore the backup to your new iPhone.

If old iPhone is no longer functional, then you would need to have an access to a backup that was made before it got broken.

There is a chance your Contacts and Messages apps where set to use iCloud to sync the data, in that case all you need to do is sign on your new iPhone to iCloud.