First notebook entry: Introducing myself

I want to learn English, and I want to teach Japanese.
Let's study with each other.(Let's study together)

First notebook entry: Introducing myself

I want to learn English,and I want to teach Japanese.
Let's study each other.

My First notebook entry Notebook Entry: Introducing myself Myself

I want to learn English,and I want to teach Japanese. / I wish to learn English and to teach Japanese in return.  (Note that I didn't use a comma before "and", because you are connecting just two sentences here. In addition to this, "wish to" is the preferable way to express your desire without being too direct and thanks to the connector "and" you can leave part of the sentence out and avoid needless repetitions.)
Let's study each other./ "I'll teach you Japanese and you can teach me English" or "Let's teach each other our respective languages" (This one is a little overwhelming perhaps.. :) ] . Here you are using "each other" as an object to the verb "study", which is mistaken. I am construing (http://www.thefreedictionary.com/construe) here, based on your 2nd sentence... 

First notebook entry: Introducing myself

I want to learn English, and I want to teach Japanese.
Let's study with each other.

First notebook entry: Introducing myself

I want to learn English, and I want to teach Japanese.
Let's study from one other.