How does one write an "extended passage of dialogue?" Is it by using colon or starting a new paragraph?
Writing dialogue doesn't have one particular format that is required, but it is important that once you establish your formatting for dialogue, you stick with it.
Here's an example, from a novel that I pulled off my shelf just now. This is The Widow of the South by Robert Hicks page 168:
"Forgive me, m'am, but I ain't really up for a lot of talking, or conversationing, or whatever it is you come to my room for."
"You're welcome to it. Put me on the next wagon out of here, I'm ready to go."
"I did not want him to leave on a wagon. Not yet."
Sometimes authors put a comma after the line of dialogue and add the character's name and even some description, like this quote from Faulkner's Light in August, page 50:
"You aint him," she says behind her fading smile, with the grave astonishment of a child.
These days there are other formats at work in stories and novels. Again, the most important thing is that you are clear in your choice of dialogue format, and that you remain consistent in its use.