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I would recommend starting with a newspaper database -- searching historical newspaper coverage will assist with determining media and public reactions. As an example, Chronicling America (1789-1963) is freely available online from the Library of Congress. You can also check into what other newspaper databases are available to you through a local academic library or local public library.
For congressional reactions, you can search the Congressional Record. The Congressional Record is the official record of proceedings and debate on the floor of the U.S. House of Representatives and the U.S. Senate. It's freely available online from 1873 to 2015 through the GPO's GovInfo website and also through subscription databases like HeinOnline.
On GovInfo, start with the 'Index to the Proceedings' in Vol. 107, Part 17 (1961) and Vol. 108, Part 18 (1962). Do a find function (Ctrl+F) search for terms like 'moon' and 'space' to get a sense of the number of (and the specific page numbers for) speeches, bills, and articles introduced to the Record on these topics.
There's also a lot of information on the space program available at the John F. Kennedy Presidential Library & Museum. They have a great blog post celebrating the 55th anniversary of the Sept. 12, 1962 speech -- and they have video and images of the address. I also see video of the May 25, 1961 address before a joint session of Congress. You can do some general searching of the site for terms like 'space program' to find more related resources.