The Pew Research Center conducts massive surveys primarily of the American public around a range of issues including politics, religion, social issues, technology and the internet, to name a few. The reports are very interesting and available to the public, but Pew also makes it's datasets available for secondary analysis. To read the reports, select the tab corresponding to the topic you are interested in (e.g. "religion"). To access the data sets, click the tab for data sets in the horizontal menu bar.
The Harvard Dataverse is a free, open access repository of data sets from research. There are, as of Fall 2018, almost 80,000 data sets included on almost any topic you can think of. Because this repository is so large, students may or may not find everything they need here, or it may be in a format that is difficult to access. However, there is so much data available, the dataverse is worth examining.
Note: the Harvard Dataverse is one of several dataverses. There are others affiliated with Universities or research centers all over the world!