Are there hard and fast rules for characters speaking to each other? As far as quotations, or without them? Are they hinged inside a paragraph or ruled to only be in a talking string?
There are three rules for conversation:
1) Indicate through some mark of punctuation that someone is speaking aloud.
This can be double quotes " , single quotes ' , dashes of varying lengths — , guillemets « , or whatever else typographic convention is in your area.
2) Make it clear who is speaking.
This is usually through some kind of dialogue tag (he said, she shouted, they whispered). You can also distinguish characters through vocabulary, grammar, diction, foreign language, accent, etc. Not every utterance necessarily needs a tag, but it should be easily and absolutely clear who is speaking any given line. If two people are talking and I have to count lines of dialogue to determine the speaker, you need more dialogue tags, variation in speech, and/or stage business.
3) New speaker gets a new paragraph.
It doesn't matter how many people are talking or how often they interrupt each other. Each new person speaking starts a new paragraph. (There are rare exceptions, but I wouldn't experiment with them until you have the basics down solid.)
With all due respect to Lauren's answer, there is a fourth rule.
4) Speech that is incidental to action stays with the paragraph that describes the action. That is, if the character runs, jumps, yells "Stop", and tackles the person they are chasing, that is one action paragraph, including the dialogue.
In prose, action usually stops for dialogue. It is too confusing to mix the two. But when words are an element of the action, they stay with the action.