First order of business, secure a new job.
Second order of business, write a professionally-worded letter of resignation that says something to the effect that regretfully, you need to move on. You don't have to give a reason in your resignation letter, but I would definitely give one verbally to your supervisor. Say it isn't working for you, it's not what you had hoped it would be, etc. Be thankful for the opportunity, and don't blame them for the job not being a good fit.
This
OP, it is YOUR life. Lead it the way you want. Don't be doing something that makes you unhappy when you have the ability to change your situation
I had the same feelings when I moved from CO to NY for a job. Stayed for a little over a year, but yearned to move back a few months in as it just wasn't what I expected. However, it was a great job with great people....just my life outside of work wasn't there. As you are finding out leases and any allowances a company may have given you to help move, or education they provide makes the situation a little more complex as in many cases, there are stipulations attached.
Keep in mind though, every action has consequences. You may want to have a recommendation from these people down the line. Also, not having a plan for what you will do after this job before you take this course of action is entirely unwise so be sure to sort that out (seems you have options but make sure.....would suck to pay 2 leases at once).
Also, a lot of industries are pretty small so try not to burn any bridges.
With that said, I would stick it out if you can but ultimately do what makes you happy. Any employer who interviews you will ask why you left so quick and will instantly have thoughts about your commitment right off the bat. Also, staying would be a benefit as you would not have to break a lease, and you would be gaining experience in an already tight job market. Plus you may not have given it a fair shot just yet.
At any rate, be professional through and through and don't feel the need to explain yourself beyond what is necessary.