Kyle attended Landmark Christian School in Peachtree City, Georgia, where he played for the school’s baseball team. During his career at Landmark, he had a .455 batting average.

In 2010, Wren enrolled at Georgia Institute of Technology, where he played college baseball for the Georgia Tech Yellow Jackets baseball team. Wren led the Atlantic Coast Conference (ACC) in hits during his freshman year, and earned the titles of All-ACC and freshman All-American during this first year. Once he became eligible for the 2012 MLB Draft after his sophomore year, the Cincinnati Reds selected Wren in the 30th round. However, he chose not to sign and instead, returned to Georgia Tech for his junior year. Before the start of the season, the National Collegiate Baseball Writers Association named Wren a preseason All-American. As a junior, Wren had a .360 batting average, and again led the ACC in hits. Following his junior year, Wren was drafted in the 8th round by the Atlanta Braves.

Before the 2013 MLB Draft, Baseball America ranked Wren as the 210th best available prospect. The Atlanta Braves selected Wren in the eighth round, with the 253rd selection. He began his professional career with the Danville Braves of the Rookie-level Appalachian League. After compiling nine hits and stealing three bases in five games for Danville, he was promoted to the Rome Braves of the Class A South Atlantic League. During his tenure with Rome, Wren had a .328 batting average and 32 stolen bases. This resulted in his second promotion to the Lynchburg Hillcats of the Class A-Advanced Carolina League at the end of the season.

Wren split the 2014 season between Lynchburg and the Mississippi Braves of the Class AA Southern League. During the first half of the season, Wren was named to the Carolina League All-Star Team. He and Kyle Waldrop shared most valuable player honors. After the season, Lynchburg named Wren their player of the year. The Braves then assigned Wren to the Arizona Fall League.

The Braves traded Wren to the Milwaukee Brewers on November 14, 2014, for minor league pitcher Zach Quintana. The Brewers assigned Wren to the Biloxi Shuckers of the Class AA Southern League in 2015. In June, the Brewers promoted him to the Colorado Springs Sky Sox of the Class AAA Pacific Coast League. Wren again split the 2016 season between the AA and AAA levels, batting a combined .322 with two home runs and 39 RBIs in 127 games. He spent all of 2017 with the Triple-A Sky Sox, batting .286 with five home runs and 62 RBIs in 128 games. He started 2018 with the Sky Sox, appearing in 47 games and batting 294 with two home runs and 28 RBIs; he was voluntarily released by the team on June 26.

On July 3, 2018, Wren signed with the Boston Red Sox, and was assigned to Triple-A Pawtucket Red Sox. He became a free agent after the season

On December 7, 2018, Wren signed a minor league contract with the Cincinnati Reds that included an invite to Major League Spring Training. He was eventually assigned to Class AAA Louisville and released on May 7, 2019.

Beginning in 2020, Wren began working as an assistant coach for the Georgia Tech Yellow Jackets, primarily focusing on hitters and outfielders. With the help of Wren, the Yellow Jackets became the best scoring offense in the ACC in 2021, capping off the season with an Atlantic Division title.