Great to hear that you have taken the first step in seeking help :) Here are some points that address your questions.
1. You can get a referral letter to see a psychiatrist from either a private GP, or the polyclinic.
2. Getting a referral letter from the polyclinic is the only way to get subsidised fees when seeing a psychiatrist within the goverment hospital.
3. Since you prefer to see a private psychiatrist, getting a referral letter from either a private GP or the polyclinic is fine. If you have a preferred psychiatrist you wish to see, you can mention this to the doctor writing your referral letter. Don't worry if you don't have a psychiatrist in mind - they will be able to recommend psychiatrists to you.
4. The polyclinic can refer you to see both a private psychiatrist, as well as a government-based one. Some polyclinics also offer psychology services and "talking therapy" by psychologists and other healthcare professionals. This team typically also comprises family physicians, nurses and medical social workers.