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At the beginning of The Hobbit, when Gandalf and Bilbo meet the first time, Bilbo says he doesn't remember his past meeting with Gandalf and only remembers him for his fireworks, but Gandalf talks like Bilbo is acting.
Do they know each other very well before? Is there any story which tells about their meeting earlier?
According to the books of the Hobbit and the Lord of the Rings, Gandalf is known to hobbits and to Bilbo, but Hobbits are rather insular in general. As an outsider Gandalf is somewhat distrusted, but he is known for his fireworks which are much liked. Gandalf from his side has taken a liking to Hobbits and knows a lot about them and their history - so is known in the Shire.
I think there is a suggestion that they did meet when Bilbo was a child, but I do not believe that Bilbo is lying or acting here - I don't think there is a personal history between him and Gandalf, I just don't think he remembers anything more than the firework displays.
That "backstory" is told in The Quest for Erebor, which is one of Tolkien's Unfinished Tales.
This is Tolkien's account of the events that led up to The Hobbit from Gandalf's point of view
A quote from Gandalf:
Somehow I had been attracted by Bilbo long before, as a child, and a young hobbit: he had not quite come of age when I had last seen him. He had stayed in my mind ever since, with his eagerness and his bright eyes, and his love of tales, and his questions about the wide world outside the Shire...
All in all the short tale
sheds light not only on Gandalf's motivations in working with Thorin and nudging Bilbo into the quest, but also reveals the dwarves' less than admirable [sic] opinions of our hobbit hero.
