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.
From my experience, beginning players (who start from not knowing how to read music at all) will play hands separately and then put them together. After you've gotten over some of the initial difficulty of just reading the music, I believe it's encouraged universally to learn/practice both at once. Of course, you can play separately to suss out some details, but on the whole, both hands should be played together. The faster you get used to playing and reading both lines-- difficult at first-- the better you will be equipped to handle more pieces that come your way. There is only a very tiny amount of piano works that are written for one hand, so really, it's in your best interest to just dive in. 

The happy medium, in my opinion, is a 90-10 split. You start out playing with both hands (and pedal, don't forget those feet!). Then from playing, you figure out what you need to work on and separate as needed. This should mostly be from the technical standpoint, because to work on the overall interpretation, you really need all the parts together and playing each part individually isn't really going to help you much in the long run. I suppose, if you were working on a piece that was in the outer reaches of your technique, the split might become more even, but even then I don't think it should reach more than 60-40. 

I'm originally a pianist, and then I got to the carillon, where similar to the organ, your feet move around a lot. It was hard for me learning an extra "line" (my feet), and it was always tempting to just practice the feet separately because it was so difficult for me and it slowed down my playing greatly. However, you move differently depending on what your hands AND feet are doing, so I became resigned to the fact that yes, no matter how excruciating, I need to play everything at once (and I do, so I'm intermediate sight-reader now). It's okay to play one at first to figure out what the melody is doing and what things you should emphasize, and just figure out the overall placement, but again, the piece is most improved when you play the entire piece, no matter how slowly at first, with all its parts and trappings.