Baer's initial training was in Cincinnati from 1876 to 1879 when he was a lithographer's apprentice at the firm of Donaldson & Co. and attended an evening modeling class taught by Louis T. Rebisso at the McMicken School of Design. It was in Cincinnati that his life-long friendships with Edward Potthast and Robert Blum were formed. From 1880 to 1884, he was enrolled at the Royal Academy, Munich, studying the antique with Stauhuber, life drawing with Gyual Benezur, and oil painting with Ludwig Loefftz. He received medals in all these classes, and gained the distinction of having a watercolor study purchased by the directors of the Academy. Following completion of his formal studies in Munich, Baer traveled to Venice and Holland with Robert Blum.
In 1885 Baer married Laura Schwenk, a German he met while attending the Royal Academy. On returning to America, the couple took up residence in the Brick Church section of East Orange, New Jersey, a community to which Baer was probably introduced when he was hired to give instruction in drawing at five dollars a lesson to an informal group of wood engravers who called themselves the Carbonari. It was in Brick Church in 1887 and 1888 that Robert Blum painted them in Two Idler's, now in the NAD collection.
Baer was a portrait painter, and gained income by illustrating for popular magazines such as Life and Century. However, he also was especially active as a teacher.
In 1892, following a year spent traveling in Europe, Baer and Blum established residence and a studio at 90 Grove Street in New York, which Bear believed to be the nucleus of the Greenwich Village art scene. That year he was principal of free hand drawing at the New York School of Applied Design for Women. In 1893, he succeeded George de Forest Brush as instructor of antique drawing at the Cooper Union, while in summer he was director of the department of drawing and painting at the Chautauqua School, Chautauqua, New York. Later, he was associated with Evelya College and Princeton University.
It was also in 1892 that through Blum, Baer received a portrait commission from Alfred Corning Clark. Clark insisted the image be rendered in miniature, thus commencing Baer's long and successful career as a miniaturist of both portrait and pictorial subjects. With Issac Josephi, Baer founded the American Society of Miniature Painters in 1899. During his lifelong association with the Society, he served as treasurer, secretary, and president. Although he never abandoned easel painting, exhibiting regularly at the Academy from 1907 to 1936, Baer was recognized as a pioneer an principle exponent of the modern school of miniature painting.
