I need a psak on a complicated (to me) halachic issue. I'm not currently affiliated with a shul, so I have no daily contact with rabbanim, and I've never needed a psak before. How does one do this? Should I just go to a local shul and ask for an appointment with the rav? Should I research who specializes in specific areas of halacha? What's the protocol?
Since many times a psak, a Jewish legal decision, is dependent upon your local circumstances, it is best to initially seek out a local, qualified Orthodox Rabbi.
Without knowing your specific location, you will need to search yourself.
If you do an internet search for your locale together with the search words "Orthodox', "Beis Din", "Beit Din", and "synagogue", you should get a contact phone number. If that fails, try calling the nearest Chabad House to you and ask if they have a phone number for the nearest Orthodox Beit Din.
Call and explain that you would like to speak with and possibly meet with the Rabbi or Rabbis for an answer to a halachic question. It may not be as complicated as you think and might be something they can answer over the telephone. Don't feel stressed about it. A Rav's life is dedicated to providing answers for those who don't know what to do in matters of halacha.
Although it isn't required, it is customary in many places to make a small monetary donation for making use of the Rabbis time and expertise like with any other professional. It's all according to ones ability to pay.