Pedantic rant:
Quotation marks should be limited to dialogue or titles or quotations of text.
Examples:
Joe said, "My favorite poem is the one that begins 'Under the spreading chestnut tree.'"
"My mom told me, 'If you use that word again inside this house you won't have a home!'"
In those sentences the speakers' words are in quotes, but since they are quoting someone (or something) else, the words they quote need to be in quotes, too. If you're already inside double quotes, anything quoted needs to be in single quotes--unless that quotation also contains a quotation, in which case you return to double quotes.
Um...we don't really need to get that complicated.
Textual quotations can be handled in more than one way. They can simply be placed between quotation marks. Alternatively--especially when they are long--they can be set as a paragraph with different margins from the main text or in a different font size from that of the main text. Typographical choices like those are usually handled by someone at a publishing house.
Titles of stories and poems and other short works are usually placed between quotation marks, while titles of books or other long works are usually italicized, but style books differ on those issues.
If you're writing fiction, using quotation marks for anything other than dialogue or titles or someone else's text is gauche. At least I can't think of any instances where it is appropriate.
Therefore, avoid:
They watched the game from the stands of fabled "Yankee Stadium."
Sometimes he could almost "sense" what his opponent was going to do next.
She became suspicious of Roderick. He was clearly "someone to watch out for."
They arrived in Paris, "the City of Light."
Some might argue that the last example is debatable, since the writer is quoting words that originated with others. However, epithets like "the City of Light" have been part of the language for eons. The quotation marks are unnecessary.
When it comes to indicating a character's thoughts, there seem to be a variety of acceptable methods, as others have noted here. When I write fiction, I rarely italicize thoughts. Depending on the wording, I find that roman type can just as effective. For me, italics--especially a lot of them--are annoying. I guess it's a matter of personal preference.
End of pedantic rant.
