Developing trust is similar to establishing a line of credit. When we first receive a credit card, our bank gives us a short leash on how much we are able to spend. This is because they are not sure whether we will pay it back.
Similarly, when we meet someone new, we typically provide superficial information. We guard our private material because we have no idea if the person is going to return it safely back to us. Eventually, if we like how they handle our superficial information, we might entrust them with something more personal. This is not a static relationship: the amount we are willing to divulge is constantly moving, dependent upon our history with the person.
This is a good way to think about the relationship that develops between you and your therapist. There is no reason to blurt out all the most embarrassing or significant material right away. It is only natural to try a normal conversation at first, and see whether that conversation opens the door to the things you came to talk about.
If not, see my post on Beginning Therapy.