Battery getting very hot and draining unusually quickly on the S2, S3, and now the S4? This is a pretty unusual problem, and I've encountered the question in various forums but never the answer. I've had this problem on and off since the Galaxy SII. What happens is I'll take the phone off the charger in the morning, and it will be extremely hot in my pocket all day, then it will die about 10 hours later. This is an intermittent problem. I thought it was loosely associated with the alarm clock, but I got a new phone yesterday because the SIII died a few months ago, and I have not had this phone for more than 24 hours and already it has the same problem. 

I haven't installed anything on it, and I've barely used any data. I have a 200 mb plan, so I can't even use data that often. The only other weird thing that's persisted is that the red arrow on the wifi and on mobile data seems to always be lit when I turn on the phone.

Has anyone seen this problem or know how to fix it?
On these phones, the behaviour you describe can be caused by a corrupted SD card filesystem. The media scanner runs all the time, trying to read the SD card.

This problem immediately suggested itself as a possibility as it would be natural to move the same SD card to your new phone, transferring the problem there. That would explain why it started right away.

You can check if this is what's happening by going into the device Settings, then Battery (on the More tab). If this is the problem, Media scanner will be at the top of the list.

Once you've confirmed the problem, take out the SD card from your phone, put it into an SD card reader in a desktop computer, and use the computer's filesystem check (i.e. fsck on GNU/Linux, ScanDisk on Windows) to repair any problems.

Even if you checked the battery stats and it wasn't the media scanner, that might give you a hint of what has been causing the problem.