There really is no answer to "what happens in a gender clinic" because each and every one is different.
I can only answer as to what I know of the gender clinics here in Australia - I believe that our gender clinics and those of the UK are more similar to those in the US (which have evolved into mostly "informed consent" clinics).
Even here in Oz, every state has a gender clinic that is different to the next. In South Australia, the process is that the GP writes you a referral to the psychiatrist who is the head of the clinic. The psychiatrist then assesses you and, if he deems you transsexual and in need of treatment, will refer you to one of the clinic doctors for hormones and, later, surgery. Basically the "clinic" is not a place, but a network of doctors who work together to treat trans* patients - a psychologist, a psychiatrist, a gynaecologist/endocrinologist and a plastic/reconstructive surgeon are who compromise the gender clinic here.
Any mental health professional can help you in theory - most people recommend you see a gender specialist, which is usually why they tend to suggest you go through a gender clinic. I would say that you have a good chance of getting help, suggestions and guidance from a gender clinic. They are there to help you with your gender issues, whether or not you're transitioning.