I would advise that your first step be to consult the archive's website, if it has one; it should detail everything that you need to know about the proper procedures for requesting and using their materials. Different archives have different procedures and rules, and it's best to be aware of them before you arrive. Don't be afraid to ask for help when you get there; archival staff maintain the facility and its records so that researchers like you can use them. If you get stuck or aren't sure about anything, tell them it's your first time using an archive and you could use some assistance.

Try to know what it is you're looking for before you arrive. There should hopefully be a catalogue that you can browse complete with rough item descriptions; identify the material you think you'll need and try to pre-order it so that it's ready for your use on arrival. This can save a lot of time and hassle for both you and the staff. Be aware of what you're allowed to bring with you - many archives will only allow you to bring so much paper into the archive room, and not all permit laptops. Many items can't be photographed or scanned, so check first if you plan to do either of these things (and you may have to pay a fee if you do).

Assuming you're working with recent or transcribed documents, your material will normally come in a box filled with folders, within which the documents will be kept. It's okay to order multiple boxes. Go through each document one at a time to see if it's relevant to your research, and makes notes on it if it is. Take notes like you would of any other primary source, but be mindful that you can't easily refer back to the source text; feel free to be generous with selecting quotes. Be comprehensive and sure to get your who, what, when, where and why down for each document you use - you might not remember key details otherwise!

Documents will normally have a reference number or code printed on them referring to their collection and place in that collection; make a note of this code for each individual document you use. You will need this to cite your findings (I have before today forgotten to do this and had to go all the way back to the archive for just 10 minutes!). Try to also identify an appropriate title for each document to make your own record keeping and citing easier. Keep the documents in the order they were presented to you; as you move through them, put them back in their proper order and return them to their folder when you're finished.

You might find this older thread helpful! The good news is archivists are generally friendly (some exceptions I'll admit...) and we are very used to people who are new to archival research, and you certainly won't be wasting our time by coming in without knowing everything. We do "orientation" classes often for students, usually at an instructor's request, but if you email the archives ahead of time and let them know you'd like a little orientation first I can't think of any university archives that wouldn't be happy to oblige.

Seriously just come in ready to follow the rules of the reading room, be nice, and we're happy to help you with the rest. :)