Yes, there definitely were touring music groups before radio. In the U.S., dating back to the early 1800s, there was the tradition of touring "minstrel shows" with white musicians performing in blackface.

Two of the more notable such groups were the Virginia Minstrels, formed in the 1840s who popularized the song "Jimmy Crack Corn", and the Christy Minstrels, led by Edwin Pearce Christy and also formed in the 1840s who popularized several songs by songwriter and "father of American music" Stephen Foster, including "Camptown Races" and "Swanee River".

Later on, in the 1880s, the vaudeville circuits began to emerge, and these venues invariably included travelling music acts at their nightly performances. Early vocalists performing on the vaudeville circuits before 1900 include Marie Cahill, Maggie Cline, Sam Lucas, Charlie Case [sic], Pauline Markham, and John Bunny.

In the first couple decades of the 1900s, before radio became a household item in the late 1920s, some of the touring vaudeville performers who sang onstage might be more familiar: Helen Hayes, Fanny Brice, Eddie Cantor, the Marx Brothers (started out solely as musicians), W.C. Handy, Maurice Chevalier, Ma Rainey, and more.

EDIT:

Re-reading your question, I didn't talk much about musical groups. There certainly were many, including the minstrel groups mentioned above. A lot of them from the 1800s would be family bands, with names like the McGibeney Family, the Barriston Sisters, and the Girard Brothers. There were also touring classical bands such as the Germania Musical Society.

Sources:

From Traveling Show to Vaudeville: Theatrical Spectacle in America, 1830–1910 by Robert M. Lewis

Four of the Three Musketeers: The Marx Brothers on Stage by Robert S. Bader