History was stored in a spreadsheet but the app on the new phone has no import feature. The app I exported from had an import/export feature which I can't find on the newly downloaded version on my new phone.
App data can be transferred from one phone to another with Helium. You can choose to backup apps with their data but that takes a lot of time and space. Better save only data and restore the apps from Play Store and then restore data from backup.
What app? If it has an import option on your old phone, the same app version should have it on the new one. Do the normal clear cache and reboot the new phone after uninstallimg then reinstalling the app. If it doesn't show up, the app developer should be contacted with your phone type and operating system.
For dummies, like myself, I start with the retail outlet for the carrier used. It's not perfect but it's fast and, then, you can concentrate on anything not transferred.