Born in East Hampton, Connecticut, Welch moved to Jonesville, Michigan in 1839 and graduated from the University of Michigan in 1846. Welch was admitted to the bar in 1847 and became a high school principal in 1849. He spent about a year participating in the California Gold Rush of 1849. Welch served as the first principal of Michigan State Normal School (now Eastern Michigan University) from 1851 to 1865. In 1859, he married Eunice P. Buckingham and they had three children.

In 1865, Welch moved to Pensacola, Florida and eventually to Jacksonville, Florida for unspecified health reasons. There, he established a lumber mill and engaged in orange growing. His wife, Eunice, died in 1867 and, in 1868, he married Mary Beaumont Dudley with whom he had two more children.

As part of the post-Civil War Reconstruction, Florida was readmitted to the United States in 1868 and Welch was elected to the U.S. Senate as a Republican. He served in the Senate for less than nine months but declined renomination, opting instead to become the first president of Iowa State Agricultural College. He served as college president from 1869 to 1883 before resigning.

From 1885 until his death, Welch served as a psychology professor at Iowa State. In 1889, he died at his summer home in Pasadena, California. Adonijah Welch is interred at Iowa State College Cemetery in Ames, Iowa.