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.
I generally only practice certain passages one hand at a time, typically when I'm having trouble with something in one of the hands. The more general rule is to isolate technical difficulties and work on them, which sometimes means working only one hand. I especially do this when I'm working out what fingering to use on a passage.

I also find that, as I'm right-handed, I have to put more time into the left hand. So, I often work with just the left hand. Chopin Nocturnes are a good example, because there are lots of leaps in the left hand that I tend to get sloppy with if I'm paying attention to the melody.

I wouldn't recommend going over just one hand in a really long passage all at once, though. It's easy to lose your sense of the music, which can make the final result sound mechanical.