If you know what radio show you want to hear, search online. There are many fan sites for particular programs or for old radio broadcasts in general.
If you just want to browse, try:
American Radio Archives
Located in Thousand Oaks, California. A very large archive of radio transcripts, recordings and books. See also:
Radio series scripts, 1930-2001: a catalog of the American Radio Archives collection. Compiled by Jeanette M. Berard and Klaudia Englund; foreword by Norman Corwin. Jefferson, N.C.: McFarland, c2006.
Firestone Library (F) Z7223.S3 T56 2006
Lexis-Nexis Academic -- under Search the News, choose Source Type = Broadcast Transcripts
America’s town meeting of the air.New York: League for Political Education, c1935-c1937.
RECAP: H1 .A565
The American forum of the air. (Washington, D.C.)
Vol. 3 (1941)-v. 17, no. 23 (June 6, 1954): Firestone Library (F) D839 .A447
Face the nation. Metuchen, N.J.: Scarecrow Press
Vol. 1 (1954)-25 (1983): Firestone Library (F) E743 .F334
New ser., v. 21 (1978)-n.s., v. 31, no. 51 (Dec. 1988): MICROFICHE 306
For February 1990 onwards, see CBS News Transcripts in Lexis-Nexis Academic.
The collected transcripts from the CBS radio and television broadcasts.
Voice of America
Transcripts on microfilm, available through interlibrary loan from CRL -- http://catalog.crl.edu/record=b1932747~S1
(1929-1991) Published by the BBC, Listener provided commentary on radio (and later television) broadcasts.
RECAP MICROFILM S01117: vols. 1-102
Microfiche 1110: vols. 103-125
Firestone Oversize HE8689 .L578q: vols. 43-117
The Museum of Broadcast Communications encyclopedia of radio. Editor, Christopher H. Sterling; consulting editor, Michael C. Keith. New York: Fitzroy Dearborn, 2004.
Engineering Library (ST) TK6544 .M84 2004
Excellent articles with bibliographies, plus a useful annotated and selective general bibliography in vol. 1. Some articles on non-U.S. broadcasting. A good starting point.
Encyclopedia of American radio, 1920-1960. Luther F. Sies. Jefferson, NC: McFarland, c2000.
Trustee Reading Room Reference (DR) PN1991.2 .S57 2000
Historical dictionary of American radio. Edited by Donald G. Godfrey and Frederic A. Leigh. Westport, Conn.: Greenwood Press, 1998.
Firestone Library (F) PN1991.2 .H57 1998
Good short articles, with bibliographies.
The encyclopedia of American radio: an A-Z guide to radio from Jack Benny to Howard Stern. Ron Lackmann. Updated ed. New York: Checkmark Books, c2000.
Firestone Library (F) PN1991.3.U6 L32 2000
Historical dictionary of old-time radio. Robert C. Reinehr, Jon D. Swartz. Lanham, Md.: Scarecrow Press, 2008.
Firestone Library (F) PN1991.3.U6 R45 2008
Radio network prime time programming, 1926-1967. Mitchell E. Shapiro. Jefferson, N.C.: McFarland, c2002.
RECAP PN1991.3.U6 S43 2002
The answer to the question "what was on the radio on Friday night in 1929?" Arranged by the day of the week and the network. (To find out more about the shows, try one of the historical dictionaries or encyclopedias.)
Radio programs, 1924-1984: a catalog of over 1800 shows. Vincent Terrace. Jefferson, N.C.: McFarland, c1999.
Firestone Library (F) PN1991.9 .T47 1999
Brief descriptions of shows. Arranged by title.
Plus books with the subject "Radio broadcasting--United States--History."