Beauty salon

Nicely done.  Here are your common errors.  Clauses connected with a coordinating conjunction must have a comma as well. Example:  I went there again today, but no one was working.   Secondly, you forget the common after adverbs starting a sentence.  Very good job!

I wanted to go a beauty salon since 5 months ago.

-> I've been wanting to go to a beauty salon since 5 months ago. 
Or: I've been wanting to go to a beauty salon for five months now.


Since my place is a countryside people can’t make a reservation in advance(It means people can’t make a reservation the day before).
-> In the countryside where I live, we can't make reservations in advance.
*It's ok not to mention the part in the parentheses. "reservation in advance" covers that
Or: In the countryside where I live, we can't make reservations a day in advance/ the day before.


I tried to go there a few times but the beauty salon is always crowded.
Or: I went there a few times, but the beauty salon is always crowded.

That is why I went home without cutting my hair.
-> But the beauty salon is always crowded. So I would come back home without getting my hair cut.




Today, I went there again. There are always over 10 people waiting for their turn but there is only 1 person today.
*Added comma and for,
person = not wrong but ok to remove



Finally I had cut my hair!
-> Finally, I got my hair cut.
Good for you. :)