Groër was born in Vienna to German parents, with whom he moved in 1929 to Czechoslovakia, where they remained for the next decade. He attended seminaries in Hollabrunn and Vienna (where he received his doctorate in theology) before being ordained to the priesthood on 12 April 1942 by Cardinal Theodor Innitzer. Groër then served as a chaplain in Petronell and Bad Vöslau until 1946, when he became Prefect of Studies at the minor seminary of Hollabrunn.