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: 
It's very normal to have issues playing with both hands at first, but if you're finding yourself completely unable to do so at all, then maybe you're attempting to play music that's too hard for you. This is a common problem for people that have significant experience on one instrument and then start another. It feels embarrassing to go all the way back to basics and play stuff that you feel like is meant for 5-year-olds, but it really has to be done. You're gonna play some Mary Had A Little Lamb, with the melody in the right hand, and single whole notes in the left. There are some method books out there that have slightly more dignified music, if that helps.

Some struggle is good (it's the only way to grow), but hitting a wall isn't productive.
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 :)