American artistic culture of the turn of the twentieth century was definitely a blend of Gilded Age American culture more broadly and of international "art world" culture. Many American artists of the nineteenth century trained abroad. Earlier in the century it most likely would have been in London, Rome, or Düsseldorf, but by the end of the century Paris was the most likely spot for an American to study art, with art academies in New York and Philadelphia also serving large groups of students. When studying abroad, an artist would be more likely to apprentice in the atelier of a single artist, such as Jean-Leon Gerome, rather than at a school.

Before I go further, however, I should say a word about the meaning of the term "American." It is difficult to name precisely one type of American artist, given the vast ethnic and geographic diversity of the country. At the same time from 1880-1910 you have international celebrity artists like John Singer Sargent, trompe l'oeil realists like William Harnett, and popular artists of Western subjects, such as Frederic Remington. My post will focus mostly on the group of artists who are most often considered "canon" in American art, namely the group of international and cosmopolitan artists who included among their ranks Sargent, Whistler, William Merritt Chase, and John La Farge. Some of what I say will apply also to artists like Harnett and Remington, though it will apply less to self-trained artists, artists who lived in the American West, and others who are outside of the main New York societal and artistic mainstream. I'm happy to answer questions about those artists if you have them.

For this group of cosmopolitan artists that includes La Farge, Chase, and Whistler, study abroad was usually the norm. La Farge and Whistler both studied in Paris, and Chase studied in both New York and Munich. Their clientele was the group of wealthy bourgeois elites centered in New York, as well as similar elites in Paris, London, and other European capitals. If you think of the sorts of people who are chronicled in the novels of Henry James and Edith Wharton, you will know the major buyers of art at the time.

In general, artists endeavored to model lives of bohemian sophistication that would appeal to the class of wealthy people who purchased their paintings. This varied on individual artist. Sargent, for example, came from Northeastern wealth, which made him a comfortable member of this group. This contributed to his ability to create sensitive and evocative portraits such sitters as [The Daughters of Edward Darley Boit] (https://upload.wikimedia.org/wikipedia/commons/9/96/The_Daughters_of_Edward_Darley_Boit%2C_John_Singer_Sargent%2C_1882_%28unfree_frame_crop%29.jpg) (1882, Museum of Fine Arts, Boston). La Farge, too, came from an international monied elite, and traveled with, among others, Henry Adams, the grandson and great-grandson of presidents.

Others had a more performative aspect to their artistic practice, one that still emphasized their worldly and bohemian nature. William Merritt Chase opened his grand studio in the famous Tenth Street Studio Building in Manhattan to visitors, allowing him to show off his large and interesting collection of imported rugs, statuary, and other exotic artifacts. Whistler developed his own sophisticated bohemian persona in London.

Artists, since they were so beholden to the financial elite of the time, generally followed the interests of those classes. The cultural historian T.J. Jackson Lears has written extensively about the condition known as neurasthenia as it relates to American cultural life around the turn of the century. Wealthy Americans of both genders began to suffer from a number of emotional and psychological symptoms that were chalked up to "over-refinement," meaning that they sought a more authentic experience of life and art. This is what prompted La Farge to accompany Henry Adams on travels to the South Seas and Japan, and also fueled the development of the Arts and Craft Movement and other attempts to return to a purer and simpler craft-based artistic practice. Others reacted to the boredom prompted by stifling wealthy society by throwing elaborate and decadent parties, such as the architect Stanford White, who was eventually murdered by the man he had cuckolded.

In short, for the top class of American artists around the turn of the century, life was generally comfortable and decadent, just like it was for the financiers and other bourgeois who patronized them. Other artists served less-opulent clienteles, and so lived less opulent lives themselves. For example, Harnett, John Peto, and other trompe l'oeil still-life artists frequently sold their works in saloons and department stores. Frederic Remington became quite wealthy through his evocation of the "Wild West," and always had to maintain a persona of tough authenticity, despite his entirely East-Coast, genteel roots.

Sources:

Lears, T.J. Jackson. No Place of Grace: Antimodernism and the Transformation of American Culture, 1880-1920.

Burns, Sarah. Inventing the Modern Artist: Art and Culture in Gilded Age America.

Lubin, David. Picturing a Nation: Art and Social Change in Nineteenth-Century America

Nemerov, Alexander. Frederic Remington and Turn-of-the-Century America