So, history as a modern scholarly discipline in the U.S. was only at its most nascent beginnings in the 1860s. In the previous decades Americans had started traveling to Germany to enroll in PhD programs there, under great German philosopher-historians such as Hegel and Herder, as well as philologists (philology is the study of the development of language, kind of a proto version of linguistics). It wouldn't be until later in the century that the first history graduate programs would be founded. At universities in the earlier part of the century, as in previous centuries, the curriculum would have been a direct outgrowth of the medieval trivium and quadrivium. Students would have primarily learned arithmetic, grammar, rhetoric, and languages (latin and greek for sure, but also German, French, and sometimes Italian). They also would have been extensively educated in the doctrines of the religious group that sponsored the college, as most colleges were religious and most college students were future ministers. Harvard, for example, was largely a training ground for Unitarian ministers in the early decades of the nineteenth century, which is why Emerson, Thoreau, and so many other of the great New England thinkers of the time were at one point Unitarian ministers.

They would have learned history, but mostly ancient history. A person interested in more recent history would have sought out the works of various historians (Prescott and Gibbon, for example) in both English and German. The same would have been true of modern philosophy.

So, by the 1860s the influence of modern German higher education was beginning to be felt in the U.S., but most curricula were still fairly traditional, relying on rote learning of the medieval liberal arts.

Sources: Diehl, Americans and German Scholarship; Winterer, Culture of Classicism

Unfortunately, I don't know much about secondary education. I could speculate, but...