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: 
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.
There are some beginning piano books that help one get the hang of integrating the two hands. Here's what a piece in such a book might look like:

A melody is presented that straddles middle C. (The hands are in the basic position where both thumbs play middle C.) As the melody goes above middle C, the right hand becomes active, but the left hand stays in position, at the ready. Within a couple of bars, the melody passes to the left hand, while the right hand rests.

There are other ways for the two hands to take turns, that was just an example.

As you progress with this approach, things will get gradually more complex.

I don't think you need to confine yourself to working with a book of this type, but it can be helpful to work with such a book alongside whatever other material you're working with. It's a relatively painless way to get both hands working together towards a common cause.

You can find books that take this approach by browsing through the beginning piano books in a music shop.