Welch was born in La Porte, Indiana but as a young boy he and his family moved West. After traversing the country, the Welch family settled on a farm near Portland, Oregon. As a student, Welch attended McMinnville College before making his way to California in 1866. After moving to San Francisco, Welch worked as a printer at the Call and Bulletin. Welch also began studying art under Virgil Williams and became an apprentice in the studio of J. W. Ogilvy.

In 1874 the talented young Welch earned a four-year scholarship to study in Europe. While in Munich, the artist attended the Royal Academy and befriended fellow artists Frank Duveneck William Merritt Chase, and John Twachtman. After leaving Germany, he moved to Paris where he continued studying art. In 1881 Welch returned to the United States and painted landscapes in the Hudson River area. Here he met his wife and artist in her own right, Ludmilla Welch. The Welchs traveled to Boston, Philadelphia, Chicago, Australia, and New York. Thaddeus and Ludmilla Welch finally settled in the San Francisco Bay area, making an artist studio out of a rustic house in the hills of Marin County. Thaddeus Welch gained acclaim in Northern California for his realistic and meticulously painted landscapes of Marin County. In 1905 Thaddeus and Ludmilla Welch moved to Santa Barbara where he continued to paint central coast landscapes. Santa Barbara was known as an artist colony of the period and the Welchs befriended many fellow artists.