You need to think about what the teacher means by "we". From the perspective of the teacher, "we" means "I and you and the other students".
When you report this, you change perspective, "I" becomes "Her", "you" becomes "I" and "the other students" is still "the other students".
So the teacher's "we" will become our "Her and I and the other students". But This is also "we" (from our perspective, it is "I and others") So you would report
The teacher told us that we were starting a new project.
Your mistake is to think about this as an exercise in mechanical substitution of words. Instead, it is a matter of understanding and reporting what happened. You must think of the meaning.
(I don't agree with the comment that this is ambiguous in any significant way. It is possible that the teacher's "we" actually means "I and some other people" in which case from your perspective it would be "Her and some other people", or "they". But this reading seems very unlikely, the teacher is addressing her class is the implied context. Reading it in any other way would be to see ambiguity where none exists.)