Data are often collected in repositories or reported in compendia and reports, which can be used as starting points on the trail to the data you need. The Statistics & Data USC Guide is also another great place to start searching for data.
When data are likely to be compiled or reported, these tools allow you to search by topic and discover data and data producers.
When data are likely to be shared by the researchers who produced them, they are likely to deposit the data in repositories. Likely candidates:
Find questionnaires, polls, and public opinion survey data from Roper, Gallup, GSS and the Census.