When you want to look up presidential speeches as well as all of a president's public utterances, then the American Presidency Project website can be useful. It's especially helpful if you are looking for dates when presidents gave their State of the Union Address or when they gave press conferences, specifically nighttime ones held in the East Room. The site is maintained by the University of California at Santa Barbara. Click on "Public Papers of the Presidents" on the upper left menu and then you can look by month or go down further on the left and search by keyword or a defined time period.
Dr. Martha Joynt Kumar is a political science professor at Towson University and has published more than 80 articles and books related to presidential communications, transitions, and the organization and operations of the executive staff. For further information on presidential transitions, communications, and contemporary history of the president's staff, visit her website.