While playing college ice hockey with the University of Denver, Moore was named to the NCAA Second All-American Team and NCHC Forward of the Year.

Moore helped lead the American Hockey League (AHL)'s Toronto Marlies to their first Calder Cup in 2018.

[2] In high school he played with the Tri-City Storm, of the United States Hockey League, in Kearney, Nebraska.

[11] During the 2019–20 season, on February 5, 2020, Moore (alongside third-round picks in 2020 and 2021) was traded by the Maple Leafs to the Los Angeles Kings in exchange for Kyle Clifford and Jack Campbell.

[citation needed] On December 15, 2022, the Kings signed Moore to a five-year contract extension with an average annual value of $4.2 million.