Archives, manuscript repositories, and special collections are not the only places to find primary resources. Papers, letters, firsthand accounts, and other primary sources of high-profile people and events are often transcribed and published. Sometimes such volumes of primary sources also contain facsimiles (that is, photographic reproductions) of the original documents.
Consider that if you are interested in John Adams, one way to get your hands on primary source material is to look at this book:
Hogan, Margaret A. and C. James Taylor, eds. My dearest friend: letters of Abigail and John Adams. Cambridge, Mass.: Belknap Press of Harvard University Press, 2007.
This is a collection of letters written by Abigail and John Adams, so these are primary sources; they've just been transcribed and put into a volume.
There are a few tricks for finding these things in the catalog. First, you have to find an appropriate subject heading. If you search the catalog by subject for Adams, John, you will find that the official subject heading we use for works by and about him is "Adams, John, 1735-1826".
Subject searching a catalog for primary sources can lead you both to archival collections and to printed anthologies of primary sources. The latter might be complete enough for your research, or might serve as a springboard for further inquiry.
Any library record will give you a list of correct subject headings that you can use to find related materials, or you can view the Library of Congress Subject Headings to get started. Complex subjects often have subheadings, each separated by dashes. These subheadings tell you even more about what the resource is about. Example subjects with subheadings about Adams, John, 1735-1826 include:
These are just a few options. You can always broaden a search by removing subheadings; some subjects have as many as four!
To find primary sources, look for these subheadings with your main subject heading:
These subject headings may not be in the second position like the examples above, but further to the right in a longer heading. You can also try using these as keywords, so you can try something like "John Adams correspondence." This method works, but it's not as precise as the method of using the exact subject heading.
This also works with topics, not just with people. For example, if your topic has to do with World War II, you'll find that the official subject heading for that is "World War, 1939-1945". If you do a subject search in the catalog, you will find that subject headings with subheadings, including the subheading "sources". Thus, "World War, 1939-1945 -- Sources" refers to works that are compiled primary sources about the war.
The one trick in this process is finding the correct subject heading for your person or topic. You can go to the catalog, do an advanced search, and choose a subject search. Plug in the name or topic, and the catalog will often tell you what the right subject heading is. Also, think big. If your topic is the Battle of Khe Sanh - the subject heading for that is "Khe Sanh, Battle of, Vietnam, 1968", by the way - don't just look for that very specific topic. Look for primary materials from people who were there. Or try searching for the Vietnam War (the subject heading for that is "Vietnam War, 1961-1975").
If you are stuck trying to figure out the subject headings for your topic, or you're just not finding many sources, come talk to a librarian. That's why we're here!