Moore's career as a mountaineer started early, with an ascent of Chimborazo and the first ascent of Sangay, in the Andes of Ecuador, in 1927.  In the early 1930s, he made the first ascents of Mount Bona and Mount Fairweather, both major Alaskan peaks, with Allen Carpé.  He also made the first unguided ascent of Mount Robson in the Canadian Rockies.

These ascents led to his most famous climb, the first ascent (with Richard Burdsall) of Minya Konka, a 7,556 metres (24,790 ft) peak in Sichuan, China. It was the highest peak climbed by Americans to date. Their small party (also including Arthur Emmons and Jack Young) also carefully surveyed the peak and settled a controversy about its height.

This was a remarkable achievement at the time, considering the height of the mountain, its remoteness, and the small size of the group. In addition, this peak was the highest summit reached by Americans until 1958 (though Americans had by that time climbed to higher non-summit points). The book written by the expedition members, "Men Against The Clouds" is a mountaineering classic.

During World War II Moore served as a consultant to the U.S. military on arctic and mountain conditions, and as a member of the Alaskan Test Expedition in 1942. In that capacity he made the third ascent of Mount McKinley.