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MeganeRei:
When I was a child, my mom took me with her when she went to the hair salon and my dad took me with him when he went to a barber. It was not for me to get my hair done when I went to either place as a child but just to bring me along as they ran their errands. I learned to not feel nervous in either environment. By my early teen years, I learned to cut my dad's hair.
My dad cut my hair until I started growing it long as a teen to add as my silent form of feminine protesting; my dad understood it. Trips to a barber shop for me were otherwise rare. The last time was for one last time when male was fading fast to my female appearance. The barber kidded me about my female hairline and told me it will go away in a few years (he did not know and I did not tell him). I had to use my male falsetto because my voice did not otherwise masculinise.
Let me tell you that if you are nervous then bring someone who has been to a barber - you are two bros getting your hair cut at their local barber man-cave - seems natural. I think you'll enjoy it - especially your first experience - if you go to an independent barber shop with a couple chairs. Those places are friendly - much as Floyd the barber in Mayberry. They will have newspapers and magazines - men's, of course. Maybe the radio or TV will have a sports event or sports talk. It will help if you can join in their conversation when the barber asks you about the local teams. Don't go overboard, just act natural.
The barber will wrap a tissue around your neck then drape an apron around your front and lap. The barber will adjust the chair upwards to his height. He may snip with scissors or use an electric hair cutter, or both. He may ask you how you like the cut as he is working - he'll turn you to face the mirror and give a hand mirror for you to check your back. When he is done cutting, stay seated as he brushes off the hair from around your neck and collar, removes the apron and tissue, and may give a few more swipes of the brush for any last clipt hairs. He'll tell you when he's done, then you can get out of the chair.
Much as women bring a picture to our salon stylist, find what you like in a newspaper, magazine, or Internet, and show it to the barber. Say something like, 'I've not had that cut for a while, I don't remember what it is called'. Some barber shops actually have a poster on the wall with pictures of various standard 'men's' cuts - especially those small shops.
Another option is a local community college cosmetology class. They have all students doing various cuts for men and women. Some classes are periodic refresher courses that some states mandate for professionals. Some do not charge at all. I had a boyfriend at a community college; he brought me in once every school quarter. I could get whatever I wanted for free but I usually accepted a basic cut; I got a perm a few times as I started passing more as female from male.
Your voice? What do your family and friends say about your voice? Go deep, I'll bet you can find your masculine voice with a little practise.
Have fun!
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