The type of resources to consult depends on the type of artist you are researching:
Researching an established practicing artist - some tips
Start with Oxford Art Online to locate information about an artist. Use the bibliography at the end of the article to identify books and articles about the artist. If you cannot find the artist, try Art Full Text for articles. If you are still not having any luck try searching for the artist in one of the newspaper databases. Finally, try the internet. Wikipedia is often useful (look at entries' reference lists), scholarly art history websites (such as Smarthistory), and many artists have their own websites.
Researching an artist established in Art History - some tips
Start with Oxford Art Online to locate information about an artist. Use the bibliography at the end of the article to identify books and articles about the artist. If you are still having trouble, search the International Bibliography of Art, Bibiography of the History of Art, and JStor for articles. If you still cannot find anything, try Dictionary of Artists or Allgemeines Kunstler-Lexicon. This provide minimal information, but often have bibliographies that are useful. Even if you do not read German, you can usually find the bibliography, if there is one, in the Kunstler-Lexicon.