One of the websites you can use to find a president you would like to hear in conversation is the one maintained by the Miller Center for Public Affairs at the University of Virginia. An example within the presidential recordings is a conversation in the Oval Office following the March on Washington. Civil rights leaders came to see President John F. Kennedy and talk to him about civil rights legislation.
You will notice there are additional topics listed on the right side of the page that give subject areas you can listen to, such as Civil Rights and Vietnam. One of the segments details the Cuban Missile Crisis. You can hear the president talk about the issues he was then dealing with, including possible ways out of the crisis. Notice that some of the conversations also include transcripts.
The site also has three of the Nixon Watergate tapes. The full collection can be found at The Nixon Library. There is one with Secretary of State Henry Kissinger giving an idea of the way in which a president – especially Nixon – likes to know how his speeches are received.