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 first started learning piano I also used to practice one hand at a time. I think it is important because then in the end it's all about coordination and if you want your hands to be well coordinated you first need each hand to know what it has to do.
Putting it all together is then a completely different story, you'll have to focus on the coordination and not on the single hand anymore plus you'll have to read 2 staff at a time (which might be tricky for someone which is not used to that), but indeed the practice that you gained playing with separate hands will help.
With experience and practice you'll do that more quickly and start putting it all together soon, however I still give a shot at each hand separately, I think it helps a lot. Of course with increasing difficulty of the piece you want to play you'll spend much more time practicing with two hands together. At times you'll have to focus on a small amount of bars when you'll have some difficult phrase (and then there are many ways which might help studying a single piece depending on the technique involved), which is also recommended when there are some particularly difficult parts in the piece (but I think this applies to the viola as well).
I cannot gauge your piano level very well, but if you are having difficulties playing both hands at the same time, then the piece is may be too difficult for your level. 

For piano players that are beyond the elementary level, I believe most players usually start learning the song both hands at the same time. This does help you get to learn the timing of the right and left hand right from the get go, instead of having to piece them together later on. Learning each separately and then piecing together later may actually be slower (depends on the piece) because of complex interweaving melodies, etc... 

However, the practice is often done separately as needed. My piano teacher used to say if you cannot play the one hand alone, you definitely cannot play it properly with two hands. When practicing one-handedly, it will often reveal mistakes that you don't really notice while playing with two hands.

For elementary players, (at least from my experience), usually the learning process is just:

* right handed songs;
* primarily right handed songs with some added notes on the bass clef
* more difficult songs with a more involved bass line for the left hand

That's why as I said earlier, perhaps if you are having difficulties of left and right handed songs, then maybe you are playing songs with a bass line that is too difficult at the moment for you.

One thing you can do to improve independent motor control with your hands is practice your left and right brain. The left brain controls the right hand and right brain for left hand. One exercise I was taught is to try rubbing your right hand on your thigh in a forward and backward motion (relative to a sitting position), and with your left hand do a upward and downward motion in a fist. Switch it up to have right hand as a fist, and left hand doing the rubbing. I know it sounds and looks retarded, but some people actually have difficulty because it involves independent directional motion.