Sometimes visiting an archive is as simple as showing up. Other times, archives require that you schedule a visit two weeks in advance and provide the exact collections and boxes that you plan to view while there. Like most of life, it's usually somewhere in between the two extremes.
A mandatory step when planning an archival visit is to check the website for their visitation policies, hours, and whether they require researchers to schedule an appointment with an archivist. Archivists are not known for tolerating chutzpah--it's what makes them so good at their jobs--and researchers must learn the rules beforehand in order to ensure a successful visit. The archive I use most frequently (the American Jewish Historical Society at the Center for Jewish History) does not ask researchers to make an appointment, which is convenient because I can set my own schedule when I visit.
Every researcher has their own "toolkit," but it's pretty standard to bring a laptop, power cord, and camera to an archive. To preserve the materials, you are not allowed to bring in bags, pens, or food and drinks. I always make sure to drink a big glass of water before my visit so I don't get too dehydrated while sitting in the climate-controlled reading room. The other important thing to know about archives before you go is that they are kept really chilly. I usually wear several layers and bring a large scarf (like a pashmina) that can double as a blanket.
When you arrive at an archive, staff ask you to put away coats and bags in a designated area. Often lockers are provided. I only bring my laptop, iPad, and scarf into the reading room. After settling down at a table, the next step is to submit a call slip for the boxes you want to view first (unless the archive asked you to pre-submit your requests). The slip usually asks for the name of the collection, the call number, and the a list of the boxes you would like them to bring. It's common for archives to place a limit on how many boxes you can call from the stacks (in my experience it is usually five). Depending on the size of the archive and how they are laid out, it can take 20-30 minutes for items to be delivered to the reading room.
Archivists are very particular about how the materials should be treated. If you are viewing books, especially older ones, they may only allow you to hold one at a time. If the binding is cracked or the book requires special care you are asked to place it in a foam cradle and to be very gentle as you turn the pages. I rarely use older books, so I confess that my description is based solely on what I've seen other people doing. I try to avoid older materials because they stress me out! Also, I like to look at more recent texts that were written on a typewriter. They're easier (and faster!) to read. It's one of the benefits of studying the twentieth century.
There are also best practices for handling papers. To avoid mixing up folders or papers, archivists ask that researchers only look through one box at a time and remove only one folder at a time. It's very important to maintain the order of the collection, so that it continues to correspond with the finding aid and so that future researchers will be able to easily locate what they are looking for.
When I open up a box, I identify the first folder that I want to look at and then carefully remove it. I note its place among the files with a marker provided by the archive (it's usually a piece of tagboard or half of an old manila file folder). I carefully place the folder on the desk and open it so that the papers are on the right side. As I go through the documents one by one, I place the read pages face down on the left. This keeps them tidy and minimizes the chances that I damage or rip the paper. When I finish with a folder, I place it back in the correct place in the box and move on, and when I finish with the box I return it to the archivists and request a new one. This process of submitting call slips, looking through boxes, and reading documents continues until the end of the archival visit.