Geetha, together with her friends, are planning to go to Darjeeling?”
Note: Prepositional phrases such as "with, together with, along with, as well as, including, in addition" do not change a subject from singular to plural. If the subject is singular, it will require a singular verb. These are not part of the subject and, therefore, have no effect on the form of the verb.

main sentence is ‘Geetha is planning to go to Darjeeling’.”
If you’d used ‘and’, then you would have said, Geeta and her friends are going to Darjeeling. OR "Geeta, along with her friends is going to Darjeeling"