A good but brief list of some web portals to start with. For a more comprehensive list, see the Art, Design, and Art History Guide:
Art 21 - "Art21 is a celebrated global leader in presenting thought-provoking and sophisticated content about contemporary art, and the go-to place to learn first-hand from the artists of our time."
Avant-garde and modernist magazines - A fantastic digital archive of Modernist magazines from 1890 - 1945. Hosted by Monoskop, a wiki for collaborative studies of the arts, media and humanities.
Directory of Artists' Book Collections: A list compiled from the library at Carnegie Mellon University of artists' book collections in the US and Canada. Some universities have online exhibits and digital collections.
Fluxus Portal - Browse online resources relating to the 1960's Fluxus Group in New York City. This portal includes links to books, videos, galleries, archives, blogs, and other movements associated with Fluxus.
International Dada Archive - Founded in 1979 as part of the Dada Archive and Research Center, the International Dada Archive is a scholarly resource for the study of the historic Dada movement. The Archive has compiled a comprehensive collection of documentation and scholarship relating to Dada. (University of Iowa Libraries)
UbuWeb - "A large web-based educational resource for avant-garde material available on the internet, founded in 1996 by poet Kenneth Goldsmith. It offers visual, concrete and sound poetry, expanding to include film and sound art mp3 archives." (from Wikipedia)
USModernist Library - "The USModernist library is the largest open digital collection of major US 20th-century architecture magazines with approximately 2.1 million downloadable pages - all free to access."
There are hundreds of museums devoted to collecting modern art and the art of today. Here are a few major museum collections in the US.
The Museum of Contemporary Art (Los Angeles)
The Museum of Contemporary Art (Chicago)
The New Museum (New York)
The Whitney (New York)
Museum of Modern Art (MoMA) (New York)
Tate Modern (London)
The Guggenheim Museums (New York)
Barnes Foundation (Philadelphia)