Where do you need to go?
Where do you need to go to?
These ones could be used at a hotel by hotel staff, asking a guest who has already asked to call a taxi, for example.
I think the first one is right. If the second one is right, why the last “to” is needed? Please let me know.
In the case of the receptionist asking a guest, the only possible question would be - Where do you want to go to? In that case the precise location would be needed.
But, Alan, that can be acceptable in an informal context similar to the one cited, I think.
In academic and competitive examinations held in India, questions of this type are set for the candidates to correct or identify or locate the errors in them.