From page 168 The Life of Mark Twain - The Middle Years 1871-1891:

On account of Livy's delicate health, the Clemenses opted to vacation in 1875 at the ocean rather than in the hills above Elmira. On July 31, with De Quille in their company, the family left Hartford for Newport, Rhode Island, some ninety miles east. They took rooms at Bateman’s Point, which De Quille described as “a sort of half-hotel, half farm-house kept by Seth Bateman, a descendent of one of the original settlers of the place.” From all indications Sam enjoyed his six-week vacation, though he later disparaged Newport in his autobiography as a “breeding place—that stud farm, so to speak, of aristocracy of the American type.” According to De Quille, Sam was "very indolent” while on holiday, though he read “about a thousand pages" of the Big Bonanza manuscript and “said it was all right.” He reportedly played baseball every afternoon with a group of boys and went bowling on an old alley in the hotel.” Along with his old friend and now disgraced former vice president Schuyler Colfax, he picnicked with the members of the chic Town and Country Club, among them Thomas Wentworth Higginson, Julia Ward Howe, and George E. Waring, an urban reformer who designed the first modern sewer systems. Maud Howe Elliott, Julia's daughter, remembered that Sam “was a tall, strongly built man, with keen hawklike nose and bushy eyebrows. His white broad-cloth dress suit, with shoes to match, always made him noticeable among a crowd of conventional evening coats. He was a brilliant, aggressive speaker, who sometimes startled his audience by jokes some thought ill-timed.” The evening of August 23 Sam spoke at the Opera House in Newport under the auspices of the Bellevue Dramatic Club on behalf of local charities. Though he had been scheduled to read excerpts from The Innocents Abroad, he read instead “How I Edited an Agricultural Paper” (1870) and “How I Escaped Being Killed in a Duel.” The Newport News reprinted his talk nearly word for word, and the Providence Journal reported that he was “the hero of the evening,” that he “delighted his hearers,” and that often during his talk the “laughter was immense.” As usual, however, he was also criticized by his cohorts for failing to be a mere humorist. “Mark Twain is a disappointment to his Newport fellow sojourners," the Boston Traveller declared, because “he does not get off so many jokes as they naturally expected."  The Clemenses returned to Hartford by September 8.

There does not seem to be place, today, known as Bateman's Point although there is a Batemen Avenue.