きこえる vs きかれる
聞こえる is used when sound is physically perceived by the ear. Think of it as a purely sonic phenomenon, i.e. sound waves reach the eardrum, resulting in the ability to hear something.
On the other hand,
聞かれる is most often used to express the passive voice of 聞く (to ask). This is used when someone has been asked something. For example, 田中さんに質問を聞かれました (I was asked a question by Tanaka). One additional usage of 聞かれる is as an honorific of 聞く, used when addressing a superior (in some dialects). This does not usually happen with the verb 聞こえる.
みえる vs みれる
As with above,
見える usually refers to the physical perceptions, i.e. the eye is in a position such that light hits the retina, resulting in the ability to see something. There are some phrases which may contain a figurative use of 見える but the standard meaning relates to being able to see something.
みれる) has several usages. One is to express the ability to view/watch - this is different than the ability to see. For example, if you only have 1 hour of spare time and the movie you want to watch is 2 hours long, you could say 時間がなくて今みられません (I can't watch it because I don' have enough time right now). Another usage is to express the passive voice of 見る - i.e. 田中さんに見られました (I was seen by Tanaka). It is possible to use 見られる as an honorific in some Japanese dialects (meaning 'view'), but it is probably more common to use ご覧になる, a special honorific verb meaning 'view'.
To summarize, when physical perceptions allow for the ability to hear or see, use 聞こえる and 見える. For the passive voice of 聞く(ask), use 聞かれる. For the ability to view/watch, use みられる. For the passive voice of 見る, use 見られる.
[Note: As pointed out in the comments, it is more common to use a different kanji for the 'view' meaning of みる. It is usually written as 観る.]