The previous responses certainly cover a great deal of what I also think should be in a high school curriculum. However, I would add to this the arts, and I also believe that physical education should be part of the curriculum.
The arts, including music, visual art, dance, and drama, are far more important to intellectual development, cultural understanding, and socialization than many realize. For example, music uses the math part of our brains, since music is a form of math, and the visual arts, too, draw on math, proportions and ratios, at the very least. All of the arts foster cultural knowledge, since art is inextricably bound up with culture. Dance involves physical movement, which has benefits I will discuss in greater detail below, and music, dance, and drama promote socialization, since they are usually the product of group effort. And of course, all of the arts nurture the creative parts of our minds, without which there would no civilization and no beauty in the world. Why wouldn't a high school curriculum want to provide all of these benefits?
The benefits of physical education as an aid to intellectual development and emotional well-being have been clearly established in study after study. On a very simple level, when we engage in some form of exercise, we get more oxygen to the brain. Our pleasure receptors are engaged when we exercise, and exercise is routinely prescribed for depression and anxiety. The maxim "A sound mind in a sound body" (Juvenal), which dates back to the first century, still holds true.
Reading, writing, math, science, and the humanities are, of course, vital in a high school curriculum, but without the arts and physical education, we are not providing all that students are entitled to and need to be successful.
To a degree, this varies depending on the particular country or type of high school, although there are certain elements that have been universal across the history of secondary education.
Perhaps the most important subjects for all school systems are reading, writing, and mathematics. Without the ability to read at a relatively advanced level, one cannot function in the modern world. Especially with the predominance of electronic communication, good writing skills are also universally important.
Bilingualism benefits both students' future careers and their intellectual development, as well as making them less insular and better prepared for an increasingly globalized world. A student who is not a native speaker of English should probably learn English as a second language, while a native English speaker should learn Arabic, Mandarin, or Spanish.
Basic mathematical skills are again essential to everything from comparing prices while shopping to running a business, and thus math is an essential part of any secondary curriculum.
For cultural understanding, world history and general humanities courses are both important, as are enough basic science courses to enable sound judgement about science-based issues such as global climate change, genetic modification, and stem cell research. A basic course on computers and internet safety is also important.
Certain non-credit practical survival courses, such as sex education, first aid, and personal finance might also be useful, but they are not part of academic studies. While physical activity, such as playing informal games and taking a walking break at the middle of the school day might be good for otherwise sedentary students, sports are generally a waste of time and money that can be better spent on academic subjects.