To save time, you can even ask for the zines you want to be waiting for you at the desk in Special Collections.
First you have to know which zines you want to read. If they are in the catalog, it's convenient to click on
It will ask you to log in -- if this is your first time, it will ask you to register (there's a tutorial at How to create a circulation request account.) -- and then you will add the zine to your list.
If you find your zine(s) on the "finding aid," it's slightly more complicated (I know, libraries, right?).