In 2011, at the age of 15, Moore was drafted in the first round of the USHL Draft. Selected eighth overall by the Tri-City Storm, the Thousand Oaks native packed his bags and headed East to Kearney, Nebraska. In the two years that Moore spent with the Storm, the hard-working and energizing winger accumulated 97 points (32-65=97) in 113 games.
At just under a point-per-game in the USHL, Moore moved up in league's the following year and attended the University of Denver on a hockey scholarship. As a freshman, Moore lead the Pioneers to a conference (NCHC) championship, while also leading the team in goals, assists and points. Then, as sophomore, Moore took another step forward and lead his team and the conference in goals (30); something his current Kings teammate Alex Iafallo would do two years later. The uptick in points and play earned Moore the conference forward of the year and a national accolade as he was named a Second Team All-American. In Moore's junior year, and final year at Denver, Moore racked up a team-high 33 assists.
Undrafted and coming out of college, Moore signed an entry-level contract with the Toronto Maple Leafs. Then, after scoring 105 (48-57=105) points in 171 games for the Maple Leafs AHL affiliate, the Toronto Marlies, the never-give-up winger got the call-up and made his NHL debut on December 23, 2018. Moore also recorded his first point in his debut against the Detroit Red Wings in route to a 5-4 overtime win.