Can I copy the app data from one phone to another phone if they both use Android? I will be buying Xiaomi Redmi Note 3 phone and my current phone is a Samsung Galaxy S3 Mini. My S3 has Android 4.2.2 and the Redmi should have Android 5 with MIUI on top as the user interface.

I'm planning to root my current phone and backup the folder where the app data is stored at (which I believe is  /data/data ). After I get my new phone, I will install all the apps I have on my current phone from Google Play. Will I be able to copy the app data folder I backed up into my new phone to restore the app settings and progress I had? If this method doesn't work, is there any way to copy my app settings to my new phone?
You can usually copy the data for user apps without issue, but there is no guarantee.  I wouldn't copy the /data/data/ folder wholesale, since it will include system app data and provides no way to easily manage it.

I recommend Titanium Backup instead.  With root, it allows you to easily backup any subset of apps (such as user apps) that you wish, and then you simply copy over the backup folder to the new phone, install it and select the backup folder, and then restore everything in a single batch operation.  I've used this to migrate apps and data all the way from Android 2.2, to 4.0, to 5.0.