Groberg was born in Idaho Falls, Idaho in 1934 and was reared there after the Great Depression. After a year at Brigham Young University, and shortly after the Korean War, he served a mission to Tonga.

After much difficulty in getting to Tonga with strikes, visa problems, and problems finding boats, he served briefly in Los Angeles, Samoa, and Fiji while waiting to get to Tonga. When he finally arrived in Tonga, his first assignment was on a remote isolated island called Niuatoputapu, which with neighboring Tafahi, was a place whose contact to the outside world was an occasional telegraph and visiting boat. On this Island he suffered mosquitoes, a typhoon, and starvation. He spent nearly a year there. While on Niuatoputapu Groberg's companion was Feki Po'uha who would later serve as district president on Niue while Groberg was president of the Tongan Mission, which at that point included Niue.

He then returned to more developed islands and served as a district president supervising smaller congregations called branches. The branches he dealt with lacked unity and morality. He had little contact with his Mission President. He nearly drowned when pushed out of a boat during a major storm and suffered exhaustion many times. He was denied an extension so he could take a group of Mormon converts to the Hamilton New Zealand Temple. He flew home and shortly after married Jean Sabin; together, they have had 11 children.

Groberg received his bachelors degree from BYU and an MBA from Indiana University. He was then involved in real estate in the Idaho Falls area. Groberg also served for five years as a bishop of a ward in Idaho Falls.

He returned to the islands of his mission ten years later as a Mission President and, since his release as Mission President, has been there many times as a Regional Representative. He is beloved by the people of that region.With much coaxing, 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. The changes he saw in Tonga from a missionary to church leader were documented in his book The Fire of Faith. He had other assignments besides Tonga as a general authority. In the mid-1990s he was Area President of the Asia Area of the church where he was closely connected with the initial sending of Mormon missionaries into Cambodia. In the early years of the 21st century he served as president of the Utah South Area of the church where he was responsible for initiating programs for missionary work among the Latino population there, and attempts to make sure that English-speaking wards hometaught the Latino members within their boundaries, even if they attended Spanish-speaking congregations. He also served as Area President of the North America West Area from August 1990 until August 1994. In this capacity in May 1992 Groberg presided over the organization of the San Francisco California East Stake, the first Tongan speaking stake of the church in United States. The North America West Area encompassed California and Hawaii.

In 2005, Groberg was given general authority emeritus status. From 2005 to 2008 he was president of the church's Idaho Falls Idaho Temple.

Groberg is an Eagle Scout and recipient of the Distinguished Eagle Scout Award.