The first diary ("my first diary" or "first diary" is a better title, sounds more natural)

I caught a cold so I slept at home. 
Someone who can speak a little Japanese whom I teach English.  I don't really understand the meaning of this sentence. Are you trying to say something like: If someone can speak a little Japanese, I would like to be taught English by them? (誰かが日本語をちょっと話せたら、英語を教えられてほしい) my Japanese is not too good, but something like that?

The first diary

I caught cold. So, I was asleep at home.(It's not "a cold", only "cold". "Cold is the reason, I was asleep.)
Someone who can speak a little Japanese whom I teach English.

(The meaning of second sentence is not clear to me. But I think you are trying to say, "If there is someone with a little knowledge to speak Japanese, I could learn English from him."