When to use 看 and 见？
In Classical Chinese, 见 describes the action that you get visual information. It implies the result, but has no duration. 见 is more like to see, 看 is more like to look (at).

In modern Chinese, you probably will not use 见 as a verb. Instead, you say 看见, which consists of a verb 看 and a particle 见. In Classical Chinese, we say 不见, but in modern Chinese, we say 看不见 (cannot see) or 没看见 (did not see) depending on the context. 见 serves as the result complement of 看 here.

The 看 in 看书 and 看电影 does not mean to perceive with the eyes. You'd better take them as collocations.

见 can be used as a verb when it means to see/meet someone, or to see something unusual and widen one's knowledge. For example, 见客人, 见世面, etc.

见面 (to meet) literally means to see someone's face. 见识 literally means to see and then to know something. But I think dictionaries will simply list them as compound verbs, because you cannot guess their idiomatic meanings from their literal meanings.

In short, 看 and 见 either have different grammatical functions (verb vs particle), or have totally different meanings.