Here is my opinion. 「〜するようにしましょう」is normally used for suggestions or when something is instructed by someone else. For example, 食べた後は歯を磨くようにしましょう。- Let's brush your teeth after eating. 毎週日曜日は掃除をするようにしましょう。- Let's do cleaning on Sundays. Can be translated to "Let's do something" and sounds like its not only one time but every particular occasion. 「〜ように話す」depends on the context but can be translated to "like" or "to" For example 彼は王様のように偉そうに話す。- He speaks arrogantly like a king. 理解できるように話す。- Speak to be understood. 「〜ようになる」"be like" "become like something/someone" 彼のようになるのが夢です。- My dream is to be like him. なるようになる。- What will be, will be ( this is a kind of idiom) There might be more ways to use but these examples came up to my mind. If you have some examples with full sentences it will be easier to explain or understand :)

「〜のようにしましょう 」は 「誰 何のようにdo」

ex. Let's speak like the teacher .

「〜のように話す」「誰々whoのように話す」

ex. He speaks like the teacher .

「〜のようになる」は become

ex. He becames like the teacher .