This year I have upgraded from a Samsung S3 to a HTC One (M7) and also had a HTC One replacement due to a bluetooth problem. In both those cases I managed the backup process myself. Having gone though this experience I think next time I will invest in the likes of Titanium Backup.
All my contacts are stored in Google contacts so that makes transition easy. Pulling down the People app left source menu you can see the contacts which are on the phone\SIM and decide if you want to move them to google.
I used the free and excellent Es File Explorer to allow me to copy backup files to a backup directory in dropbox and restore from there on the new phone. The dropbox folder location is found at /mnt/sdcard/Android/data/com.dropbox.android/files/scratch/
I use SwipePad as my launcher. It has an option to back up via the More Spaces add-on (allows larger grids to show more apps on the screen). The backup is stored in /storage/emulated/0/swipepad_backup.db. I copied this to DropBox and restored from there to this location, this was picked up on the new phone.
I use Go SMS Pro for my SMSs. The app has a backup & restore option via the More Tools menu, you need to copy to /sdcard/GOSMS/backup/localbr directory to restore.
WhatsApp has a Backup conversations option via the Chat Settings sub menu. This creates a proprietary timestamped .crypt file in /storage/emulated/0/WhatsApp/Databases. However they have no restore or merge option (if you have been using another phone for a while) You can include the backup file in the WhatsApp Databases folder after downloading, on install it finds this and asks you to restore it, however this restore completely failed for me
I use Podkicker for podcasts, it has a backup within the app with location given /storage/emulated/0/Android/data/com.podkicker/files/podkicker_backup.opml. If you copy the backup to this location it is picked up.
I use LLama to automatically make my phone go into quiet mode at work, etc. It has a backup option in the app which copies 5 files to /storage/emulated/0/llama. Restoring the files to this location works. You them have to go into the Recent tab and re-hookup cells to an Area – minimal effort.