Make sure your old iPhone is logged in in the Dawn Patrol app. You can log in using your email address or using Facebook. Log in with the same account on your new iPhone, and your data will automatically synchronize.
If you don’t want to log in you can simply restore a backup of your old phone to your new phone, and your Dawn Patrol data will be transfered.
Do you want to make sure your Health and Activity data also is transfered to your new phone? Switch on “Health” app date in iCloud settings on your iPhone or use iTunes backups. (see Apple’s support page)