Robson has said he wants to become a professional chess player, and his parents hoped for him to gain a chess scholarship to college. In April 2005, at the "Super Nationals" (the world's largest scholastic chess tournament) in Nashville, Tennessee, he won every match he played and emerged as the national champion in the elementary age (K-6) division. By winning this title, he earned a four-year scholarship covering full tuition and fees, along with a housing stipend, to the University of Texas at Dallas. The scholarship has a cash value of about $48,000 to non-Texas residents. The only stipulation is that the winner must meet the university's entrance requirements at matriculation. In 2009 Robson was the recipient of the Samford Fellowship. In early 2012, Robson decided to attend Webster University instead of UT Dallas.
Robson has won seven national scholastic titles (including regulation events and blitz events). In addition, he has represented the United States in international scholastic events since 2004. Robson finished in the top ten at each of the World Youth Chess Championships from 2004 to 2007, and he tied for first place in the U12 section of the Pan American Youth Chess Festival in 2005 and 2006, taking the silver medal on tiebreak on both occasions.
Robson was awarded the title of FIDE Master (FM) in June 2005 after tying for first place at the Pan American Youth Festival in Brazil. He earned the USCF title of National Master (NM) in January 2006 by raising his Elo rating above 2300 (the minimum required for the title). Robson earned the three norms required for the IM title in only six weeks: the first at the 6th North American FIDE Invitational on November 3, 2007, in Chicago, Illinois; the second on November 27 at the World Youth Championships in Antalya, Turkey, and the third and final norm on December 10 at the University of Texas at Dallas (UTD) GM Invitational in Dallas, Texas, making him the youngest IM-elect in the United States, beating the previous record-holder Hikaru Nakamura by one month.