There's a huge variety of things they can offer, from different types of therapy, different activities, some time limited, some are not. Its definitely worth asking questions because then you can make an informed decision.
I think its normal to feel embarassed and probably a bit awkward. The therapist or leader will be used to dealing with a variety of actions and circumstances of people entering therapy and will have seen how you feel many times before (if that makes sense). Its important to remember that you will be in control and can say as much or as little as you like. Obviously the more they know the more they can help, but equally, you have control over what you say.
There are some sorts of therapies that don't involve talking and opening up as you may be thinking. Counselling requires it, but things like CBT (Cognitive Behavioural Therapy) and DBT (Dialectic Behavioural Therapy) are focused on teaching skills to help yo cope with whatever is going on in your life, so that's more focused on dealing with the present, not healing the past.