Welcome to Android Central! Do you know if the contacts on the old phone were saved to your Google account, or only the local Phone account? Go to contacts.google.com on your desktop browser -- do you see the contacts there? If so, all you need to do with the new phone is to log in with the same Google account, and those contacts will sync automatically. This is one of the major advantages of saving your contacts to your Google account.
If, however, you saved them to the Phone account, then you may be out of luck, since contacts in that account don't sync anywhere automatically. It's up to you to make periodic backups, usually with 3rd party backup software. If you never backed them up, then I'm afraid you'll have a difficult time retrieving them. If they're very important, you may want to spring for a screen replacement.