How do I transfer app data/settings from an S3 to an S6?
Since your S3 is rooted. You could make app backups with a 3rd party app, Such as Titanium Backup, but to use Titanium back up to restore them on your S6, It would need to be rooted as well. I know Google claims that its back up service saves app data and settings as well. But I think that only works if the app dev has added in the option to store things in the cloud. 

Personally I can say that TB works like a charm for me. For non-rooted options(I assume your S6 isnt rooted if you dont have access to a pc). It looks like https://play.google.com/store/apps/details?id=com.koushikdutta.backup.license&amp;hl=en can back up to the cloud. Only reason I linked the paid one is cause the free one doesnt allow you to back up to the cloud.