It’s easy to get tripped up when writing dialogue. In North America, the rule is to use double quotation marks around the words of the speaker; this seems simple enough, but the most common problem lies in the placement of punctuation other than quotation marks. The standard rule is to include periods and commas within the quotation marks:
She said, "I don’t believe you are telling the truth."
"I don’t believe you are telling the truth," she said.
"I don’t believe you," she said, "and I no longer trust you."
"I don’t believe you!" she exclaimed.
"Should I believe you?" she asked.
"Fine, don’t believe me," he replied. "You’ve never trusted me."
Note that dialogue tags (she said, he replied) must be a "speaking action," whereas non-speaking actions (he snorted, she glared) are not punctuated as tags and should be treated as separate sentences:
"I don’t believe you are telling the truth." She glared at him.
"Fine, don’t believe me." He snorted with disgust.
Use single quotation marks to mark dialogue within dialogue:
"And then I looked at him and said, ‘I don’t believe you!’" she said.
She said, "And then I looked at him and said, ‘I don’t believe you!’"
"And then I looked at him and said, ‘I don’t believe you!’ and he walked away."
Another general rule is to start a new paragraph for each new speaker. It makes for a clearer picture of the give and take of a "conversation" and helps the reader switch from speaker to speaker.
Some writers use italics to set off dialogue, while others use no special punctuation at all. However, unless you’re Hemingway or Joyce, it’s best to save the italics for quoted thoughts and use traditional punctuation for your dialogue. For clarity’s sake, whatever format you choose, keep it consistent, and your readers will thank you.