I would recommend completing the module Apex Basics and Database on Trailhead. Two of the units in that module cover how to query and access data across relationships.
However, in this case, you don't need to traverse a relationship at all.
appointment.Doctor__c.Id
should simply be replaced by
appointment.Doctor__c
The relationship field itself contains the related record Id.
This code is at risk of a number of governor limits problems (in particular the Visualforce view state limit), as well as general poor performance, due to:
- executing and storing unfiltered queries, which return all records in the database.
- performing filtering in Apex instead of appropriately filtering SOQL queries to show relevant data.
I would recommend you consider removing your unfiltered queries and rebuilding updateDoctor() to perform a new SOQL query when the selected doctor changes to acquire the doctor record and their appointments, using a parent-child relationship query (covered in the module linked above). You would then be able to remove updateAppointments() entirely.