Do I have to complete the DUI class to get my Driver's License back in California or just begin enrollment in the DUI class?

If this is a first dui you have to begin the classes to get your license back on a restricted basis. You must ultimately finish the classes and submit proof to the dmv to get the restriction (to and from work, in the course and scope of work and to the dui program) removed. If you drop out of the classes you will not get your license back until you finish the classes.