I know that if you have a quote within a quote you use single quote punctuation, like this:
"Grandmother was a nice lady, 'Have a great day!' she'd tell me."
But what if you have two speakers? Do you start a new paragraph? Example:
"Grandmother was a nice lady, 'Have a great day!' she'd tell me. Dad wasn't as nice, 'you're a wimp' was all he'd ever say."
Depends on how you wanted to do it. You could have it as one continuous quote (since it's one person speaking), like you have it, although your punctuation is a bit wrong.
It should be:
"Grandmother was a nice lady. 'Have a great day!' she'd tell me. Dad wasn't as nice. 'You're a wimp' was all he'd ever say."
But if you want to separate it out into multiple paragraphs, for instance going into a bunch of other ways Grandmother was nice and how Dad wasn't quite so nice, you'd do it similarly, but don't close the quote until your speaker is done, but start each paragraph with open quotes. Only close once the whole speech is done:
"Grandmother was a nice lady. 'Have a nice day!' she'd tell me. She'd also make my favorites cookies for me every time I visited. 'Sweets for my sweet. Don't worry about finishing these. I will always make more for you.' And she always did.
"Dad wasn't as nice. 'You're a wimp' was all he'd ever say. He was a real jerk. 'Boy, you want a whipping?' 'Stop your whining, you sissy.' 'You gonna amount to nothin'.' Those were his favorite things to say to me when I was growing up."