Drew went to Lowndes High School in Valdosta, Georgia. He played college baseball at Florida State University under head coach Mike Martin, as did both of his brothers, where he was Baseball America's Freshman of the Year, and ACC Rookie of the Year.
He is the younger brother of Red Sox outfielder J.D. Drew and former major leaguer Tim Drew. The Drew brothers are the only trio of first-round siblings ever selected in the Major League Baseball Draft.
Drew was selected to the 2006 All-Star Futures Game.
In a game on September 1, 2008, Drew hit for the cycle against the St. Louis Cardinals, becoming the first player to do so at Chase Field. Batting leadoff, Drew singled in the first inning, tripled in the third inning, and homered in the fifth inning against St. Louis Cardinals starter Joel Pineiro. He added a ground-rule double in the seventh inning against reliever Kyle McClellan. It was the fourth time a player hit for the cycle in the 2008 season. On the same day, Adrián Beltré of the Seattle Mariners hit for the cycle. This was the first time that two players hit for the cycle on the same day since 1920.