How can I restore all my backed up data? I wiped my old phone and got a new one today, and on my old phone I had all of Google's back-up options turned on, but it didn't sync any of my app data when I installed the apps on my new phone.

Just in case something like this happened, I also copied and pasted the files of my old phone into a folder on my PC. However, nothing changed when I copied all those files onto my new phone. What am I doing wrong?
Copying and pasting folders from the old phone to the new one doesn't link it to the app installed in the new phone. Instead, try backup apps such as Titanium Backup, and make a complete backup of all the apps which you require from the old phone. Now, a folder named TitaniumBackup would be created in your SDcard. Copy that folder and paste it in the new phone. Install Titanium Backup in the new phone and try to recover the apps with data. Your issue will be resolved.