てご覧(らん)(Please) try to, (Please) look

ないでください is a polite way to ask someone not to do something. It's neutral and appropriate for strangers or people of equal status. In contrast, てごらん is a gentle request form often used by adults to children, etc., that adds a nuance of 'Give this a try' or 'Try doing this (to see how you like it).'

The dictionary/non-past form of a verb + な expresses a blunt command not to do something. In contrast, てごらん is a gentle encouragement to try doing something. For example, 食べるな！ means 'Don't eat it!' while 食べてごらん means 'Try eating it' Both are used toward someone of lower status, such as an adult speaking to a child.

てください is a basic way to express a polite request ('Please do…'). For example, 食べてください would mean 'Please eat (this).' It is polite and neutral, and can be used with strangers or people equal in status to you. てごらん is a gentle request form often used by adults to children, etc., that adds a nuance of 'Give this a try' or 'Try doing this (to see how you like it).'

てみる expresses trying something to see what it's like, similar to the English 'try (doing)', e.g. 食べてみる ('I'll try eating it.'). てごらん is a more directive form that encourages someone else to try something, e.g. 食べてごらん ('Try eating it.' as a gentle request). てごらん is often used by adults to children, or superiors to subordinates. Among peers, something like 食べてみて would be more common.

Both are command forms, but with different tones. なさい is a polite but direct command, often used by teachers or parents, e.g. 食べなさい ('Eat.'). てごらん is softer and gently encourages someone to give something a try, e.g. 食べてごらん ('Try eating it'). Both are typically used from adults to children, or superiors to subordinates.