About my working holiday

Firstly I stayed in Brisbane, which was near my language school.
I lived in homestay for the first month.

This could be written as one paragraph

For the first month I stayed in a homestay in Brisbane. It was near to the language school that I was attending.

Hi Smayu (and Mark)

A few comments if I may.

Firstly, I stayed in Brisbane, which is near where my language school was.

or:  I went to Brisbane where my language school was.

- I would omit Firstly altogether ("firstly" sounds a bit odd to me so I never use it, preferring "at first"), it really means nothing here, unless you say something like you stayed in Brisbane at first then moved to Sydney or another city after that.

- Brisbane is a city which is much bigger than the school, you do not say the city is near the school.  See Mark's suggestion for the paragraph.

My English was bad and I was in a beginner's class during the time I was a student for the whole three months.

or:  My English was bad and I was in a beginner's class during the time I was a student the whole time.

The sentence is correct but sounds rather awkward, from the context it's already understood that you were a student at the language school, so I want to suggest something that would make the sentence sound more natural.  It's just a writing style.

However,  I could communicate with my friends, so I was satisfied, except I couldn't get any nice jobs.

Watching many English programs on NHK, the Japanese TV station, I notice that Japanese speakers tend to overuse the word "however" and almost always place it at the beginning of a sentence.  Again, it's not incorrect, it just often makes the sentence too strong and seems awkward.  I would try to say it in another way.