Groth early decided to become an artist. He studied--probably very briefly--at the school of the Art Institute of Chicago, and then developed his own skills by obsessive sketching. His first sustained employment came in about 1933, as art director of the fledgling Esquire magazine. When he and Esquire parted company in the late 1930s, Groth went to New York to try to work as a painter. It would have been at this time that he attended the Art Students League, where he worked under Arnold Blanch, Robert Brackman, and George Grosz. A naturally romantic, colorful character, Groth came into his own when on assignment for the Chicago Sun, he went to Europe as an artist-correspondent during World War II. His first report was also probably his most dramatic: the liberation of Paris, filed from the vantage point of having been riding in the first American army jeep to enter the city. One of his latest World War II stories was of entering Berlin with the Soviet army. The first of his books was Studio Europe, recounting and illustrating his wartime experiences.
Following the war, Groth returned to freelance illustration work, contributing to Life, Look, Esquire, and Colliers, among other popular magazines. He was especially employed in covering a wide variety of sports, from bull fighting to baseball for Sports Afield, Argosy. In 1950 he became the first art director of the new Sunday supplement magazine, Parade, and it was for that publication that he again became a war correspondent in Korea, and then in French Indo-China. His second book, Studio Asia resulted from these experiences. He would cover American military action twice more, going to the Belgian Congo in 1961, under auspices of the United States Air Force, and to Viet Nam in 1967, for the United States Marine Corps.
Groth's illustrations were rendered in dense, nervous ink line, and watercolor, and it was in the watercolorist classification that he was elected to the Academy. From 1942 he taught a weekly night class at the Art Students League. He also taught at various times at the Pratt Institute, and at the Parsons School of Design, both in New York. Groth served on the faculty of the Academy school from 1962 into 1965.