I was more intimidated by the receptionists and the waiting room than the actual counselling sessions. I still am to this day and this is because I trust the therapists will be knowledgable, experienced and understanding with regards to my issues, but I don't have such expectations of receptionists etc.
I've had more counsellors than I can count on one hand, and on the whole they didn't help me much. I think I either need a really domineering counsellor who knows the exact answers and the right way to make me change, or a psychotherapist to go into the deep dark void with, perhaps to do regression or hypnotherapy, you know more serious stuff.
I've tried group therapy for social anxiety and it was rubbish. Everyone has SA for different reasons and for a lot of people it is a symptom of a bigger problem. I have been later diagnosed by a psychiatrist to be a complex case with personality disorders (and a complex early childhood), so to solve my issues within a group environment where we just tackled the surface did not work and isn't going to work again. Also some people there were working and clearly articulate and chatty, whereas people like me couldn't even talk to people and had intense anxiety facing anybody whether they had the same problem or not. But perhaps some people might find it helpful if all they have is something bordering on shyness.
About the actual session, they usually start by asking about the general nature of your problems. They want to know about your whole life history. This is where it's useful to tell about anything which you feel would be relevant in causing your problems. For instance my parents were absent during my childhood with a mother who left me with different carers. Because I didn't know what normality was like in a family setting I didn't mention that and wish I had as i've realised how important that was. Mention family relations, early childhood, friendships, major events. Usually the therapist will lead the questions and if you don't feel comfortable talking about something just tell them so. They won't pressure you to say anything. I am guessing that your therapist will be lightweight (a psychologist). A psychotherapist is a heavyweight who will go deeper with you. A psychiatrist prescribes you meds and diagnose conditions.
I used to always cry on my first sessions because it felt overwhelming to talk about all my issues, but I have grown more detached over time.