According to this brief description:
When an old lady buys a giant mushroom, a wind carries them off on an
adventurous trip.
The author is Michael Jennings or Michael G. Jennings. He also wrote a second book in the series called "Mattie Fritts and the Cuckoo Caper. "The Flying Mushroom" was published January 1st, 1973 by Harper-Collins Publishers.
In it, a lady goes out to the shops to buy mushrooms and ends up with an extremely large one, about the size of herself.
He looked at her shrewdly/ "Oh, yeah? I know your kind—always complaining. I bet I could show you the biggest mushroom in the the world and you wouldn't buy it"
"I would so!"
"Sold! Stay right here. It's in the back.
He returned in a moment with a huge mushroom over his shoulder. Its stem was about six inches across, and at least a yard long.
It's a windy day, and the mushroom acts as a sail and carries her up into the sky.
"I declare!" Mattie exclaimed, as a gust of the almost-forgotten wind threatened to topple her. "We can't even go to Grand Central with King Mushroom here. We'd best take him home before we make any plans."
But when she tried to turn the corner, she found that she couldn't. Bending forward, with the giant vegetable held before her, headed into the wind, she ran in place for a few minutes but could make no headway.
"We must give this a good think, Daisy," she muttered. "Perhaps if I turn the mushroom the other way, I can back down the street.
It was the wrong thing to do. A sudden gust made the top of the mushroom billow like a parachute, and she had to run with it to keep from having it torn from her grasp. She hadn't realized she could run so fast and so effortlessly until she turned her head and saw that her feet were off the ground, making running motions in space.
She lands on a rock in a bay (I want to say it's San Francisco Bay) which turns out to be a giant talking octopus who befriends her.
"I declare!" Mattie exclaimed. "We've landed on a tiny island. No wonder we didn't see it. The rock is almost the same color as the water. Anyway, there's room to sit." Shouldering King Mushroom like a large parasol, she made herself at home.
"Well, now, this isn't at all unpleasant, is it?" she addressed her companions. "It's a nice morning to sit out on the water in the sun and wait for help to arrive. Actually, Daisy," she said to her hat, "this is more comfortable sitting than those benches in Grand Central."
The island shifted under her and a voice said, "I shouldn't be surprised. It's my head you're sitting on."
"Oh, goodness," Mattie said. "I don't remember whether I've ever talked to an island before, but if I did it's certainly the first time one ever spoke to me."
"I'm not an island," the island said. "I'm an octopus."
She then flies away together with the octopus on the mushroom.
In no time at all they were airborne again, Mattie settled comfortably into the crook of a tentacle.
Later on they pick up a tiny tortoise (who is a girl with a name that sounds like a boy's)...
Pete squeaked his loudest. "I said I'd better clear up one misunderstanding. I'm not a he, I'm a she. I'm a lady turtle."
...and a massive woolly dog named Woof, who were both pets in the same family.
Well, I'll be dogged," Woof said. "That's great! Got room for a hitchhiker, Mattie? I'm not very heavy. It's all fur, or hair, or something."
After flying most of the way across the United States, they find the family to which the tortoise and the dog belonged. The father of the family was a man we saw in San Francisco very early on in the story, and the mother is living on the other side of the country with three kids, one of whom is called Jason.
Jason was only a year older than Dana, and of the same size and coloring. Strangers took them for twins, and even some relatives got them mixed up. That annoyed both of them.
"That sure is a funny-looking horse that's over-taking Rutabaga," said Marc in the self-assured voice of an older brother.
One scene that stands out in my mind is when the octopus pretends to be a car, with the lady on top and the mushroom as a steering wheel, and they go to a gas station to get water.
The young attendant stepped briskly from his office, and started to salute as a vehicle rolled, or rather wobbled, up to his pumps. The salute changed to a scratch of his head when he got a look at the strange contraption.
Octopus had become an automobile, more or less. Mattie sat back against one upraised tentacle. Four others had been coiled into lopsided wheels and he swayed into the station under their power. Two tentacles were contorted to resemble gearshift and parking brake levers. The spare one looked like a tire mounted at the rear. His eyes were open as wide as possible, to serve as headlights.
Towards the end of the book, the lady wins a horse race while riding the dog.
"Who finally won the race?" Dana asked. "We did," said Mattie. "We're disqualified for the prize, of course, because we weren't entered, and Woof isn't a two-year-old horse."
"He's not any kind of horse," Dana said.
"They're not convinced of that," said Woof. "The veterinarians out there are still arguing."
"In any case," said Mattie, "Woof, Pete and I were all declared honorary winners."
