In Japanese there are words which mean "to try something or attempt at doing something", e.g. 試す, 試みる and つとめる.
However, at the same time we also have two grammatical constructions "verb＋て＋みる" and "verb＋(よ)う＋とする".
In English we just say "try (to do) something" or "attempt (to do) something", but in Japanese it is not that simple. So what is the difference between all these different expressions? Precisely when is each used?
There is a fairly big difference in meaning between 「～～てみる」 and 「～～ようとする」 that makes them virtually noninterchangeable for the better speakers/writers.
「～～てみる」 means "to try something out (usually for the first time) (to find out how it is, how you like it, etc.)".
whereas
「～～ようとする」 means "to attempt to ~~ to achieve some kind of goal (however insignificant it may be)". It is highly volitional in meaning.
The difference would become even clearer if one used actual words in front and turned the phrases in question into their past tense forms.
「フグを[食]{た}べてみた。」 vs. 「フグを食べようとした。」 フグ = a poisonous blow fish
If you say 「フグを食べてみた。」, you actually ate it. If you say, however, 「フグを食べようとした。」, we do not know if you actually ate it. This should already be a huge enough difference.
「フグを食べてみた。」 means that you tried it out because, for instance, you had heard so much about how yummy fugu was and you had been very curious to find out how it tasted.
Thus, this sentence would often be followed by a sentence explaining your impression of the experience such as 「とてもおいしかった」、「あまりおいしくなかった」, etc.
「フグを食べようとした。」 means that you actually intended to eat it but with this sentence alone, we do not know if you actually ate it. You probably went to a fugu restaurant with all your good intentions of trying the fish but you may have backed off at the last minute fearing you might not return home alive. Thus, this sentence would often be followed by a sentence telling whether you actually ate it or not such as 「でも、こわくて食べられなかった。」、「食べてみた。おいしかった。」, etc.
Finally, to group all the verb phrases mentioned in your question for meaning:
Group A: ～～てみる、[試]{ため}す、[試]{こころ}みる
Group B: ～～ようとする、つとめる
The difference between "verb + しようとする" and "verb + してみる" has been already answered. In short, the former is "try to do something," and the latter is "(actually) try doing something."
～を試す is used with various kinds of nouns. I don't know how to combine this with another verb (although you can say "試しに + verb + してみる"). It means "test/check" how something works/tastes/looks/etc. The 漢語 version of this is 試験(する), which sounds more scientific/technical.
- 新しいソフトウェアを試す。さっき作った料理の味を試す。数学の能力を試す。
- 刺激を与えて反応を試す。 apply stimulus and check the reaction.
～を試みる is combined with nouns which can be followed by "する". The meaning of this depends on the context and the noun used with 試みる. Let's see the following examples:
#1 質問に回答しようとした。 made an attempt to answer the question (maybe in vain)
#2. 質問に回答してみた。 (actually) tried answering the question.
#3. 質問に回答を試みた。
Now you know the big difference between #1 and #2? Sentence #3 usually is a formal equivalent of #2. But you can still use it in the same sense as #1. (質問に回答を試みたが、いい言葉が思いつかなかったので、やめた。)
Other context-dependent examples:
- 世界征服を試みる make an attempt to conquer the world (rather than "actually conquer the world as a test")
- その問題の検討を試みる (actually) investigate the issue (not "try to investigate")
- 対話を試みる try to communicate / (actually) try communicating (not "check the communication")
- 反応を試みる try to react / react as a test (not "check the reaction")
Note that these nouns can be used with する: 征服する, 検討する, 対話する, 反応する.
(noun)に努める is an explicit and formal way to say "work hard / make an effort at something."
