In the first chapter of The Hobbit, "An Unexpected Party", Gandalf (speaking to Thorin) says,
Thrain your father went away on the twenty-first of April, a hundred years ago last Thursday, and has never been seen by you since.
We're also told that Bilbo tended to forget things
unless he put them down on his Engagement Tablet: like this: Gandalf Tea Wednesday.
It appears to follow, then, that the Unexpected Party occurred on the Wednesday after Thursday April 21st; that is, on Wednesday April 27th. The departure, of course, occurred the next day: Thursday, April 28th.
Note: Tolkien himself did not, at least in the text of The Hobbit, pay close attention to this date (one of the characteristics that distinguishes the book from his other fiction). In Chapter 16, "A Thief in the Night", the narrator states that Bilbo
drew from a pocket in his old jacket (which he still wore over his mail), crumpled and much folded, Thorin's letter that had been put under the clock on his mantelpiece in May!
This could be considered as counting against the simple interpretation of the first chapter, or it could be simply an authorial error.