How do I deal with the overwhelming ambiguity of my project? I have a bachelor degree in applied mathematics and I'm currently doing a PhD in molecular genetics as a computational biologist. I started in my thesis lab in January and since then I've had serious problems understanding exactly what my supervisors expect me to do. 

Here are a few of the underlying issues:


I've had to transition from a first principles based approach to problem solving to a top down, data-oriented worldview - something that has not come easily.
There are no upper-year students doing a project similar to mine that I can shadow while I learn the ropes.
My supervisors talk about my project and direction at a very high-level, so the details of what I should be doing or how I should be doing them are unclear.
I face a constant stream of discouragement, be it my peers succeeding while I don't, my supervisors just expecting me to 'figure it out' when it comes to the details, my unsupportive family, etc... and it's affecting my ability to work.
I did a number of very technical projects in my undergrad and I enjoyed them immensely. I have little love for my current project - likely a symptom of not succeeding.


How do I deal with this? Is it a normal part of the process to feel lost and hopeless - even this early on?

Most importantly: At what point should I cut my losses and drop out or switch departments?
Is it a normal part of the process to feel lost and hopeless - even this early on?


Yes, it is very normal. Particularly early on. There are many things you do not know or understand yet. I spent at least a year of my Ph.D. on projects that never amounted to anything (literally), and know there were many times I asked myself these same questions.

Now to the more practical question: 


  How do I deal with this?


Based on my very similar Ph.D. experience, my best advice is to "own it". You need to convince yourself that you are doing the research that will go in your dissertation. It will take you personally making mistakes and learning from them. It will take you developing a workflow where you quickly prototype ideas so that the bad ones can be discarded without too much time lost. I would bet that although you feel like you are treading water, you are developing skills and insights that will be useful once you gain more traction. 

For this to work, you may need to stop expecting your supervisors' directions. Instead, systematically "try stuff" on your own. Start by addressing some simple concrete question related to your project: try something naive and see why it breaks down. After you identify the flaws, think about and implement some naive modifications. Iterate on this process and document everything carefully. Use your supervisors as reviewers -- explain to them what you did and ask for feedback rather than direction. In my case, iterating this method led to solving a problem parallel to the one my supervisor had original posed, but in a more 'hot' area of research.

Finally, regarding this point: 


  I face a constant stream of discouragement, be it my peers succeeding while I don't, my supervisors just expecting me to 'figure it out' when it comes to the details, my unsupportive family, etc... and it's affecting my ability to work.


I often felt deeply discouraged as well. First, comparing yourself to your peers is a recipe for frustration. Avoid this at all costs (easier said than done, I understand). Second, I found exercising, pursuing my hobbies, and meditating all helped curb the day-to-day frustration of the "treading water" feeling.  Talking to other grad students, particularly from different disciplines, also helped. I found this feeling is far more common than some of my peers would let on -- and being able to open up to those a bit removed from your specific project was always helpful.