The Little Red Riding Hood in Into the Woods is clearly taken straight from the pages of the Grimm Brothers version of the tale. There are any number of events and lines that are almost identical, yet in the first line of the Grimm Brothers's tale we find the first difference. The Grimm Brothers describe the girl as charming, whereas Sondheim and Lapine created a character who was anything but.
The story starts with the girl's mother loading her up with cakes and wine, which of course means she doesn't have to go to the baker. Red's visit to the baker in Into the Woods was a necessary addition which has no influence on the story. In the Grimm version her mother advises her to "look straight ahead of you like a good little girl and don't stray from the path". Notably, Red does promise to obey her mother's wishes.
When Red finally comes across the wolf, the Grimm brothers make a point of noting that she knows nothing of the dangerous nature of wolves. This is reflected in Into the Woods when she suggests to the baker that he was worse than the wolf. This is because in her limited experience, he was worse than the wolf. He tried to steal from her, whereas the wolf was friendly. At this point Little Red has been cunningly seduced by the wolf. Seduction is a theme that comes from the early oral versions of Little Red Riding Hood. In one version, the wolf even asked the girl to take off all her clothes one by one and get into bed with him. This theme was removed by the Brothers Grimm but is hinted at in Into the Woods especially when Cinderella's Prince later seduces the baker's wife, and it reminds us of the wolf. To see this idea taken further, have a look at Putting It Together, a Sondheim revue where Hello, Little Girl is taken entirely out of context, and without changing a word, becomes about a predatory older man, (presumably named Mr. Woolf).
The song Hello, Little Girl barely contains an original idea, yet the humour of it certainly does not come from the Grimm Brothers. The flowers, the birds, the fresh bouquet and the wolf's desire to have two meals in one day, are all straight from the story. One thing that can not be traced to Grimm is the suggestion of the wolf that there may be another path worth exploring. This is obviously Sondheim expanding on the life metaphor: It is important to stay on a path and know where you're going, but get the hint that your path may not necessarily be the beaten track.
Other ideas that appear in both versions of the story are the three large oak trees, the uncomfortable feeling she gets, the idea that the prettier the flower, the farther it will be from the path, the idea of filling the wolf's belly with stones and of course the list of things that Little Red notices aren't quite granny like about the wolf.
Some things in Into the Woods are either original, or don't come from any source I've found. The idea of Little Red eating the sweets before she gets there is unique. As is of course the idea of someone trying to take her cape, and her manner in general.