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Best Bets (Non-Database):
General Social Survey (GSS)From the publisher: "For five decades, the General Social Survey (GSS) has studied the growing complexity of American society. It is the only full-probability, personal-interview survey designed to monitor changes in both social characteristics and attitudes currently being conducted in the United States."
Pew Research CenterFrom the publisher: "Pew Research Center is a nonpartisan fact tank that informs the public about the issues, attitudes and trends shaping the world. We conduct public opinion polling, demographic research, content analysis and other data-driven social science research. We do not take policy positions."
American National Election Studies (ANES)From the publisher: "To serve the research needs of social scientists, teachers, students, policy makers and journalists, the ANES produces high quality data from its own surveys on voting, public opinion, and political participation. Central to this mission is the active involvement of the ANES research community in all phases of the project."
Census Survey ExplorerAllows you to search for specific Census Bureau surveys/censuses based on your needed criteria
Data.govFrom the publisher: "Data.gov is the United States government's open data website. It provides access to datasets published by agencies across the federal government. Data.gov is intended to provide access to government open data to the public, achieve agency missions, drive innovation, fuel economic activity, and uphold the ideals of an open and transparent government."