From where I am from, it is considered that doctors in government hospitals are the best in the country in terms of experience and brilliance. Since these are people who get admissions to these medical courses based on educational merit,.
First, figure out who the doctor is that you want to consult and see if they have a private practice. Some of these doctors have their consultations at home after their working hours. See if you can go that route; it might be easier to reach them that way.
Not all doctors are good bhai. It depends on the department you want to visit.
Getting appointment is difficult without knowing someone (either a politician or a doctor who has known contacts here or some other govt servant). But that's also subjective, some departments are overloaded while some have comparatively lesser waiting times.
The experience without any connection is pathetic in pgi. Never been to aiims so no idea.
No idea why everyone praising AIIMS here, get to real fact, you will either need a resource or stay in line for long hours to even get first appointment. After the appointment, in case of surgery, you have to wait for months depending upon the case history and resources available.
If the patient issue is complicated, get it treated by private hospital like Max or Fortis.
Getting appointment isn't a problem there but getting treatment is.
Long ques, over utilised capacity of patients, dynamic behaviour of staff, negligence, carelessness, etc etc. are some of characteristics.
Don't forget to post your experience.