Begg was born in Glasgow, Scotland in 1952 and still carries a distinctive accent after nearly 30 years of ministry in the United States. On November 2, 1972, his mother died. Regarding the incident, Begg wrote that "more spiritual progress is made through failure and tears than success and laughter." Begg graduated from The London School of Theology in 1975 and then served eight years in Scotland at Charlotte Baptist Chapel in Edinburgh and Hamilton Baptist Church in Lanarkshire. Begg is a Council member of the Alliance of Confessing Evangelicals. He was educated at Trent University and Westminster Seminary. Among Begg's influences are: Martyn Lloyd-Jones, John Stott, Derek Prime, Eric Alexander, Sinclair Ferguson, Dick Lucas, Eric Liddell, Charles Haddon Spurgeon, Jim Elliot, and Hudson Taylor.

Begg's wife, Susan, is American by birth and Begg himself became a U.S. citizen in 2004. He and his wife have been married over 30 years and have three grown children. In 2004, he played Stewart Maiden in the film Bobby Jones: Stroke of Genius.