JSTOR is a very rich source of material on music in general and there is a great deal on Shostakovich and the quartets here. Reviews on each of the quartets and more.
There are so many avenues for finding material at a large university library. But Google Scholar can be extremely helpful. If you search in Russian you will find a large number of articles. There are also very useful Russian webistes such as http://magazines.russ.ru that will provide access to the full text of many of the literary-cultural journals, many with full text for the last 15 years.
There is a website that provides an overview on each of the 15 quartets, Shostakovich: the string quartets at http://www.quartets.de/index.html. There are also numerous avenues for locating reviews of individual performances: any one of the databases on music, Google Scholar, online newspaper archives, such as the Historical Archive of the New York Times (UIUC database).
The article below originally appeared in New York Magazine, May 24, 1982 and has been posted in its entirety on Google Books. While there is a somewhat limited selection of materials on Google Books for Shostakovich it is surprising the works that turn up there and it should not be overlooked if you do not have access to a large research collection.
There are a wealth of sources available through most libraries that will make finding material on Shostakovich fairly easy. There is one thing to keep in mind. Cyrillic alphabet words are rendered in a variety of ways into latin characters. See "A Few Considerations" for a list of possible spellings.
The number of results you retrieve from a search will be radically affected by the spelling you use. Some of the most useful"subscription" databases include: