Woodbury received a BS from Utah State University in 1952, his master's degree from Brigham Young University in 1953, and completed his PhD at the University of Illinois in 1954. Soon after, Woodbury was hired by BYU as an Associate Professor of speech, a position he held from 1954 to 1961. In 1965 Woodbury returned to BYU as a professor in the Theatre and Media Arts Department.
Woodbury wrote extensively, publishing over 50 works, among them books, monographs, articles, and book reviews. He also was invited to lecture on the arts throughout Europe and Asia.
Additionally, Woodbury served as Chairman for the Utah Alliance for Arts Education from 1976-1985, Vice President of the University and College Theatre Association of America from 1982-1983, Chairman of the National Committee on Royalties and Leasing from 1972-1981, and as President of the Rocky Mountain Theatre Conference in 1959. He also served on the Board of Directors for Eagle Systems International, the Utah Arts Festival Advisory Council, and the Repertory Dance Theatre from 1977-1979.
During his career at BYU Woodbury was given many honors including the dedication of a professional production of Oedipus Rex in 1985 by the Theatre Borgia of San Francisco. He was also named the Fine Arts Guild Patron of the Year in 1982, received the BYU Alumni Distinguished Service Award in 1975, and was awarded the Sidney B. Sperry Symposium Award in 1975. In 2009 he was further honored by the College of Fine Arts and Communications and received the Franklin S. Harris Award and was also invited to speak at the college's August convocation.
In 1992 Woodbury retired from BYU and was granted emeritus status.
- The Deseret News - Lael J. Woodbury (29 September 2013)
- College of Fine Arts and Communications Convocation Speech (14 Aug 2009)
- "Hello, in There" BYU Devotional Address (16 July 1974)
- "Continually Before the Lord" Commissioner's Lecture Series
- Address given to Faculty and Staff of the College of Fine Arts and Communications, (24 Aug 2005)