Question: I've started to teach myself how to play the piano, and one of the difficulties I've found is trying to get both hands to do things at once. I either forget one hand entirely, or have one hand trying to mimic the other. 

This reminded me of many years ago when I was taught to play the viola. For many weeks, we did not use the bow, instead focusing on getting the left hand to do the things it was supposed to, and learning to read music. In hindsight, this makes sense to me, as it's often important to get used to some fundamentals before attempting to do everything at once. 

So I was wondering, when learning the piano, is it better to learn each hand on their own, or muddle through the difficulties of synchronizing them from the beginning? Or, if there are arguments for both methods, what are the pros/cons of each? What problems may develop, or what difficulties might be avoided? Or is there perhaps a happy medium that has proven the most successful?

For my personal musical background, I have about nine years' experience with the viola, though I haven't played it in about five years (stopped after high school). I also play the guitar, but never learned to read music for it.

Answer: 
When I was taking piano lessons I learned each piece three times: Once with the right hand, once with the left hand, and the third time putting both together. After a while, the third "learning" came a lot more quickly, and after a few years I would start to slowly sight read both parts together.

Today I will still practice the hard pieces one handed from time to time, especially when the timing is really disconnected between the two hands and I need at least one to go on automatic pilot. There are some pieces (*Leyenda* by Albeniz comes to mind) where the two parts are so intertwined it never makes sense to try to learn them alone. Usually that's clear from the beginning when it's the case.
That depends on the piece sometimes. My old teacher used to say practice one separately until you've got it down then "see" them together. Honestly, I'd recommend practicing one measure at a time by repeating one until you've got it and then move on. Lastly, play from the beginning after a few measures to smooth over any rough spots :)