Norris graduated from Goddard High School in Goddard, Kansas in 2007. At Goddard High, Norris played third base before transitioning to catcher, and also won a Class 6A Championship title.RISE Magazine named Norris its 2006–2007 Kansas Baseball Player of the Year. He committed to attend Wichita State University on a baseball scholarship.

The Washington Nationals selected Norris in the fourth round of the 2007 Major League Baseball draft. Norris spent the 2007 season with the GCL Nationals, Washington's affiliate in the rookie-level Gulf Coast League. He played for the Vermont Lake Monsters of the New York–Penn League in 2008, the Hagerstown Suns of the Class-A South Atlantic League in 2009, the Potomac Nationals of the Class-A Advanced Carolina League in 2010, and the Harrisburg Senators of the Class-AA Eastern League in 2011. Baseball America rated Norris the 38th best prospect in baseball prior to the 2010 season and the 72nd best prospect in baseball prior to the 2011 season. He was also chosen as the Nationals' second best prospect prior to the 2011 season.