I have this awful habit of postponing things uselessly, in fear. Fear of what? Nothing. I'm not always fearful. I even like it, sometimes. But I'm afraid of "something". Then I tend to do things better when I'm:
1. feeling pressured to do it right here, right now, like when working in class with the teacher and all, or
2. feeling I have someone supporting me and expecting/hoping success of me, like going shopping, which I never do alone, but love to do with someone, or
3. in a busy state..
I think number 3 is your best bet. When I'm in "work mode", I want to do all sorts of useful things that I've been postponing. Or rather, I tell myself that I won't remember them later, or they'll be too intimidating later, and that now is the time. For example, when on the bus or walking home from school, I have this strong urge to do my important phone calls. Don't be afraid to do it while buying some goods, or whatever. It will feel like much less of a big deal if you're choosing pasta sauce at the same time.
If that won't work, I know you're reluctant to having others support you, but 1. and 2. work great.
For the appointment itself, I suggest you book for a longer time (they'll charge you for a brushing instead of a haircut, which isn't a huge amount of extra money), bring a few haircuts you'd maybe like (print pictures or drawings, or even draw them yourself), and ask them to help you choose carefully.
Oh, and from experience. Don't go for that nice girl without a diploma who's nice but has strong opinions. If it happens like it did to me, she'll try to influence you and/or be stubborn in doing mostly what she wants, and it'll end up bad. The busy/not too talkative kind aren't the best either, because those "just do what the customer asked for", acting a bit like machines, in an effort not to insult you. But what you need isn't to be just done what you want. It's your first "real" haircut and you're no expert.
Even if it's a bit more expensive, I think you should aim for someone with a real expertise who will truly take the time to help you choose the best haircut for you, and explain their suggestions properly. That's more or less what I'll be trying to do when I get a haircut.