I'd been to a few therapists before seeing my current therapist, none of them were very good experiences and I was wary of therapy. But my therapist now knew what I was seeing her for and specializes in transgender patients so it went really smooth. If your therapist doesn't know why you're there, they'll ask, so just tell them you're having gender issues and they'll start asking questions.. just answer them honestly and you'll be fine. A lot of people, myself included, weren't looking forward to having to see a therapist, but they really are there to help you. It also helps a lot if you like your therapist. If you don't like the first one you meet with, if it's at all possible, it might be a good idea to meet with someone else. I know not everyone has options though, so if that's not possible, just stick it out, it's really not too bad and going forward with transition will be worth it.
I see you asked about cost as well, that completely depends on insurance. I think my therapist's sessions cost about $120 for one hour, but I only have to pay a $20 co-pay and insurance covers the rest