If we're going by the exact wording in the book, it seems pretty clear that the decisions and choices are largely Gandalf's.
Gandalf only reveals his reasons for choosing Bilbo when he feels unjustly criticized during the dinner and meeting at Bilbo's home; he states that "There is a lot more in him than you guess, and a deal more than he has any idea of himself." He also admits (if it can be considered an admission, since Gandalf clearly feels no guilt over it) that he placed the mark on Bilbo's door that identified him as a burglar looking for work, and that he expects everyone, including Bilbo, to go along with the letter of his actions; specifically, that they asked Gandalf to find a fourteenth member, and Gandalf has chosen Bilbo, who is a burglar. There is to be no further questioning of whether Bilbo is actually a burglar, or whether he had a choice in the matter.
Bilbo does seemingly attempt a passive-aggressive avoidance of his new duties by sleeping in and dawdling about the house, until Gandalf comes to check on him. From that point, Bilbo says he simply doesn't know why he went with the dwarves, particularly without any of his usual preparation or baggage. Bilbo bemoans his circumstances and wishes he were back home on more than one occasion, but he never abandons the group.
It could be said that Bilbo had some element of choice in the matter, particularly by keeping his end of the bargain (even if he didn't agree to it in the first place) but I think it was largely Gandalf who threw Bilbo into the quest and let the details sort themselves out later.