How did it happen in the movie?
After Gandalf leaves, we get a bunch of shots of Frodo and Sam walking and walking… and walking. In between walks, they see a bunch of Elves travel to the West at night.
Next we see Frodo and Sam cross a corn field, where they accidentally bump into Merry and Pippin, carrying some stolen vegetables. It appears they are being pursued by one angry farmer Maggot, owner of said vegetables, and make a run for it. In doing so, they end up on the Road again, where Frodo gets an eerie feeling just in time and makes them all hide under a tree root. Just in time it appears, because seconds later a spooky Black Rider appears and starts sniffing ominously.
After their narrow escape, the four friends are running for their lives in the woods and Merry leads them to Bucklebury Ferry, where they escape said Rider for the second time.

How did it happen in the book?
In the movie it seems that the journey to Bree doesn’t take up more than a couple of days. In the book, however, it takes up a few months: there is the move to Crickollow (as described in A Conspiracy Unmasked), the journey through The Old Forest and the stop at the Barrow-downs. All of this has been removed from the movie, in favour of a shorter and less descriptive journey.

In the book chapter ‘A Shortcut to Mushrooms’, Frodo, Sam and Pippin are travelling to Crickhollow together. They wake up after their nightly encounter with Gildor-Inglorion, and debate the Elves and the presence and identity of the Black Rider they encountered the day before.
Frodo then decides to cut through the country and take a shortcut to Crickhollow, especially to avoid passing a certain pub. On their journey they stumble onto Farmer Maggot’s estate. Frodo is afraid of the farmer and his dogs because he once stole mushrooms there when he was a young lad, but Pippin knows the farmer rather well and as a matter of fact they are greeted most kindly. The farmer offers them a copious meal, and tells them about the stranger who came to his door and asked about ‘Baggins’.
After the dinner, Maggot offers to drive them to Bucklebury Ferry in his cart, but on the way they meet Merry who had grown a little worried while waiting in the new house with Fatty Bolger.
The rest of the journey is quite uneventful.

As you can see, Jackson put some echoes of this chapter back into the movie: the good farmer gets a cameo as the reaper persuing Merry and Pippin, and it’s Merry that leads the Hobbit to Bucklebury Ferry.

Mistakes
– Apparently, there’s a car driving by when Frodo and Sam are standing in the cornfield: it was spotted by several people in the theatre. It’s nowhere to be seen on the DVD version however, and there is some speculation of it being a chimney or something of the kind. Mysterious, indeed.
– Some people have complained about the fact that the cornfield in this scene is not old-fashioned English corn, but ‘maïze’. They figure maize wouldn’t have existed in Middle-earth, since this is a representation of the Old World, and maize was first discovered in America by the conquistadores. However, ‘The Two Towers’ contains the famous discussion about how to eat a rabit, where Sam suggests it should be eaten with ‘taters’. Potatoes, too, were discovered along with the ‘New World’, so it seems some Tolkien fans are more consistent than the good professor himself. So no, it’s not a mistake.
– According to the book, both Merry and Pippin were good friends of the farmer and his wife, and regular guests. So there’s no way they would be stealing his crops.

Borrowed Lines
– Pippin’s comment about what it was exactly that they were doing, is actually the title from this chapter: ‘A shortcut to mushrooms’.

Bookie Details
– As said above, the good farmer has a very small cameo part as the reaper swinging above the corn.
– It’s Merry that leads the Hobbits to Bucklebury Ferry, which is a small tribute to Merry’s roots – as a Brandybuck – lying in Buckland.