Groberg was born in Idaho Falls, Idaho in 1934 and was reared there after the Great Depression. After a year at Brigham Young University (BYU), and shortly after the Korean War, he served a mission to Tonga. Groberg received a bachelors degree from BYU and an MBA from Indiana University. Groberg also served for five years as a bishop in Idaho Falls. He later returned to the Tongan islands as the mission president and later as a regional representative.
He wrote a book about his mission from his memoirs called In the Eye of the Storm. His missionary experiences in Tonga became the subject of the Disney film The Other Side of Heaven. In April 1976, he was called to be a general authority of the LDS Church. In May 1992, Groberg presided over the organization of the San Francisco California East Stake, the first Tongan speaking stake of the church in the United States. In 2005, Groberg was given general authority emeritus status.