Buxton started playing baseball when he was six years old. He attended Appling County High School in Baxley, Georgia, where he played for the school's baseball, basketball and American football teams. He played quarterback and wide receiver for the football team. He stopped playing basketball during his sophomore year, but played football through his senior season. In addition to playing the outfield, Buxton also served as a pitcher for Appling County.
Scouts noticed Buxton when he was 15 years old. He was recruited to play on a traveling summer team. He appeared in baseball showcases, including the East Coast Pro Showcase and Under Armour Baseball All-America Game. Buxton is considered a five-tool player. MLB.com rated Buxton the best prospect available in the 2012 Major League Baseball Draft.
The Minnesota Twins selected Buxton with the second overall pick in the 2012 Major League Baseball Draft. Buxton signed a $6,000,000 contract with the Twins on June 12, 2012. He made his professional debut in Minor League Baseball (MiLB) with the Gulf Coast Twins of the Rookie-level Gulf Coast League (GCL), where he batted .216 in 26 games. Later in the year, he was promoted to the Elizabethton Twins of the Rookie-level Appalachian League, and batted .286 in 21 games. He was named the top prospect in the GCL.
Buxton started the 2013 season with the Cedar Rapids Kernels of the Class A Midwest League. After he batted .340 with eight home runs and 32 stolen bases to start the season, the Twins promoted Buxton to the Fort Myers Miracle of the Class A-Advanced Florida State League in June. By midseason, MLB.com, Keith Law of ESPN.com, and Baseball America viewed Buxton as the best prospect in baseball. He represented the Twins at the All-Star Futures Game. After the season, Buxton was named the Midwest League most valuable player. Baseball America named Buxton the Minor League Player of the Year, and he won the J. G. Taylor Spink Award, given jointly by Topps and MiLB, as their Player of the Year.