日本語はやすいじゃありません。
日本語は、かんたんじゃありません。
Easy to do is しやすい, but やすい alone is not used.

～やすい is a verb suffix; it goes on the end of a verb and has the meaning “easy to ~”.

• 話しやすい easy to speak
• 読みやすい easy to read

Using it on it’s own (without the verb) is incorrect.
Both the first and second users changed it to the adjective かんたん which just means “easy”.

The second also changed じゃありません to ではありません.
じゃ is a contraction of では. They mean the same thing, but では is less colloquial and more preferred in writing. Either way is fine, as evidenced by the first user leaving じゃ in place.

日本語をよくはなしたい。
日本語をうまくはなせるようになりたい。
うまく＝good at ｜ はなす(speak)＋できる(able)→はなせる ｜ なる(become)＋たい(want)→なりたい

Although よく (from the adjective よい) can mean “well”, it often carries the meaning of “often” or “frequently”.

• すしをよくたべる I often eat sushi. I eat sushi well.

The user changed よく for うまく (from the adjective うまい) which means “skillful” or “great”.

The bigger change that the user made is from はなしたい to 話せるようになりたい.
There are a few bits of grammar here. I’m not sure what order Textfugu goes; you may not have covered them all yet.
At the end of the phrase is ～たい which I think you understand (since you used it) means “want to do ~”.
The ～たい is attached to the verb なる (なりたい) which means “to become”. So far we have “want to become”.

The ～ように is probably the hardest part to understand (and is worth looking into on its own).
A basic translation would be something like “in the way of　~” or “in such a way that ~”.
It is frequently paired with なる to mean “to reach the point that ~”
So ～ようになりたい means “to want to reach the point that”

話せる is the potential form (look into this too, it’s useful) of 話す and means “to be able to speak”.
Sooooooooo, 話せるようになりたい means “want to reach the point that I am able to speak”.

Which is really what you meant to say when you said “I want to speak Japanese”. Sometime Japanese requires you to be more literal and explicit.

The second user correct this as 話すことができるようになりたい.
You see that the ようになりたい is the same as above.
The only difference is that 話せる is replaced with 話すことができる.
These mean exactly the same thing.

今、動詞と”ために”をべんきょうします
今、動詞と「ために」をべんきょうしています。
今 and しています go together.

今、動詞をべんきょうします means “Now, I study verbs.”
今、動詞をべんきょうしています means “Now, I am studying verbs.”

You can see why the second is better.
You probably just haven’t gotten to the point where you learn the ～て form of verbs and the pressent progressive tense yet. You will.