Frodo knows Gandalf because Gandalf was like Bilbo (Frodo's grand-uncle) guardian on Bilbo's adventures. Now, Frodo knows Gandalf through Bilbo.
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Frodo is actually sitting a ways back in the cart from Gandalf, but because the seats they are on are not actually shown and compared, and the side of the cart is the same, the audience gets the effect that Frodo is just smaller, and not actually sitting back.
He made him leave the Shire with Frodo.
frodo bagins the hobbit FAMILY FEUD: frodo gandalf gollum legolas aragorn sam gimli
They were close to him, and knew that something was afoot, but not knowing what that was, they had Sam spy on him. (In the book) In the movies, they knew because Sam was roped into it without a choice by Gandalf, and Merry and Pipin knew only beause they ran into him, and desired to help when the saw that danger was following him.
Frodo did volunteer. It was Gandalf's belief that the ring chose him, and possibly knew it would destroy him to take it away. Also, Gandalf had a high regard for Hobbits, and that they could resist it's temptation longer than any of the other races, because of their lack of desire for power.