Use double quotation marks for almost everything. This includes:
For direct speech and quotes:
"What happens next is up for discussion," she said.
When you want to draw attention to a particular word or phrase:
Avoid writing "eg" when you can use "for example" instead.
To refer to an individual song, article within a larger publication or short story:
My favourite short story is "Don't Look Now" by Daphne Du Maurier.
If the quote is more than one paragraph, add double quote marks at the beginning of each new paragraph, but only at the end of the final one:
The Vice Chancellor said: "The university is committed to making a difference in all our cities and regions.
"We are stewards of a pioneering and entrepreneurial tradition of creativity and innovation.
"Our aim is to empower and support students and staff to collaborate in order to solve problems and improve lives."
Use a colon to introduce reported speech:
She said: "Our vision is to be a university without borders and to change the world for the better."
Any punctuation relating to the quote goes within the quote marks, and any other punctuation sits outside it:
“Out,” said Lady Macbeth, “damn’d spot!”
“After all, tomorrow,” said Scarlett, “is another day.”
For quotes within a quote we use single quotation marks:
“When I think of it,” he said, “I have always liked to hear ‘I love you’ as often as possible.”