What are the early 20th century's literary movements and trends?

You also would benefit to look beyond modernism to postmodernism, which in many ways rejects some of the key tenets of modernism and "plays" with literature and genre in often highly amusing and different ways. One excellent example is the author David Mitchell, who writes postmodern fiction that really "breaks the mould" of what we have come to call a novel - Cloud Atlas is one of my personal favourites.

Along with the Modernist movement was Minimalism, a movement in which the work is stripped down to its most fundamental features.  The aim of Minimalism is to allow the reader to examine the work more intensely without the distraction of composition, theme, etc. Raymond Carver is a 20th century author who employs this technique. 

Modernism was probably the most dominant literary movement or trend in the early 20th Century.  The advancement of a doctrine that stressed the lack of a unifying order or principle helps to take greater shape and definable form with the commencement and conclusion of World War I.  The lack of faith in structure and order and the embrace of a world set in motion through chaos and an accompanying sense of disillusion helped to drive much of the literature and thought of the early 20th Century.  I think that modernism and its expression in writers and works of the time period must help to make it a significant literary trend of the time, helping to shape the paradigm for the century.

What were the major literary genres and trends in early 20th-century America?

Naturalism was a major American literary genre at the turn of the twentieth century. Naturalism is considered an offshoot of realism, a late nineteenth-century movement which sought to depict life without romantic flourishes or sentimentality. Naturalism is distinguished from realism in its pessimism regarding human nature. The naturalists possess a deterministic view of life in which people are controlled by hereditary forces or their environment. Stephen Crane, Frank Norris, and Jack London are the three most prominent naturalist authors.

Naturalism petered out around the 1910s, and by that time, modernism began to flourish on the American literary scene. Modernism largely arose from the chaos left in the wake of World War I, in which the values of the nineteenth century were obliterated. Modernist literature breaks with traditional modes of storytelling, experimenting with techniques such as stream-of-consciousness narration and fragmentation. It tends to focus on a sense of loss as well, reflecting the disillusionment many Americans experienced in the wake of the Great War. F. Scott Fitzgerald, Ernest Hemingway, Langston Hughes, and Elizabeth Bishop were major writers of the American modernist movement.

What were the major literary genres and trends in early 20th-century Britain?

The beginning of the twentieth century was a vibrant period of change in the British literary world. The novel remained a highly important genre. Writers like Arnold Bennett continued to develop the realism of the nineteenth century, piling on detail to put a reader in a scene while relying on omniscient narrators to create a sense of objectivity. However, this method was increasingly challenged by writers like James Joyce and Virginia Woolf, who believed that focusing on the interiority and subjectivity of personal experience more accurately captured reality than omniscient, all-knowing narration. How can we honestly purport to know how the universe operates, these writers asked? How can the godlike assuredness of an omniscient narrator reflect reality?

Poetry likewise was beginning to change, moving from the regular rhyme schemes of poets like Arnold and Tennyson to more broken, experimental forms and from a generalized optimism to a more pessimistic view of what seemed, even before World War I, a fragmenting reality. Poetry, like the novel a dominant genre, also became more subjective, as illustrated in T. S. Eliot's 1911 "The Love Song of J. Alfred Prufrock." As World War I and its aftermath shocked and traumatized the British, poetry reflected this disillusion and alienation in the antiwar poetry of Wilfred Owen and Siegfried Sassoon, and after the war in poems such as Eliot's "The Wasteland" and Yeats's "The Second Coming."

The despair, alienation, and experimentation of early twentieth-century British novels and poetry produced a new and powerful body of literature as writers came to grips with a less secure reality.