見る, 見える, and 見せる: looking, being visible, and showing
Distinguish a deliberate act of looking from something being visible and from showing something to another person. Track what を, が, and に mark in each sentence.
見る: deliberately look at or watch something
見る presents an action in which someone directs their eyes toward something. 私は映画を見ます。 Here, 映画 is what the person watches, so it takes を. テレビを見る and 写真を見る follow the same pattern.
毎朝、窓から山を見ます。 describes a habit of looking at the mountain through the window. The focus is on what the person chooses to look at, rather than on whether the mountain happens to be visible.
見える: be within sight
見える describes something being visible or coming into view, and the visible thing normally takes が. この席から舞台がよく見えます。 means that the stage can be seen well from this seat.
暗くて字が見えません。 says that the characters are not visible because it is dark. 見える is not the regular potential conjugation of 見る; it focuses on visual accessibility rather than a deliberate act. Depending on context, 見られる can instead present an opportunity or ability to see something.
見せる: show something to someone
見せる means causing something to be seen by another person. 私は友だちに旅行の写真を見せました。 Here, 友だちに marks the viewer, while 写真を marks the thing shown.
先生に作文を見せます。 expresses an intentional act of showing an essay to the teacher. Do not read 見せる as being able to see, and do not reverse the roles marked by に and を.
Quick check: In この席から舞台がよく見えます, what is emphasized rather than a deliberate act of watching the stage?
Show the answer and why
The sentence emphasizes that the stage is clearly visible from this seat. That is why it uses 舞台が with 見えます. It is not 見せます, which would mean that someone shows the stage to another person.