In Modals^x, Punctuation^x

Hi,I am encountering difficulty in distinguishing whether there should be a pair of quotation marks around for a case like this. I feel, a lot of times, people skip the word 'that' or do not care to put quotation marks if what is noted requires the punctuation marks. He said you should...

I'm taking a course to become an English Teacher in Spain and I have to write a summary of all the lessons I attend during the course. Once I have all the summaries I'll have to hand them in to my teacher and I wouldn't like to have any grammar mistakes. Could you take a look and check everything...