～てみるTo try to

〜ようと思う・〜おうと思う expresses intention, similar to 'I'm thinking of doing X,' e.g. 行こうと思います ('I plan to go.') てみる conveys the idea of trying something to see what it's like, e.g. 行ってみる ('I'll go and see.'). These forms can be combined, e.g. 行ってみようと思います ('I plan to go and see what it's like.')

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 can be translated in English as 'try (doing)', but differ in nuance. ようにする means to make an effort or form a habit, e.g. 毎朝走るようにしている ('I try to run every morning.') has the nuance of making a point of doing so. In contrast, てみる means to try something out to see the result, e.g. 走ってみた ('I tried running.'), which has the nuance of doing so to see what it was like.

Verb[volitional]とする and てみる are both translated as 'try', but the former implies attempting something that may not succeed, e.g. 話そうとした ('I tried to speak.'), which implies that maybe the attempt was unsuccessful. In contrast,てみる focuses on trying something to see the result, as in 話してみた ('I tried speaking.'), which has the nuance of doing so to see what it was like.