I was raised pretty gender neutral (but with a full awareness of what sex I was, duh). We lived in the country and i think in some ways it's easier there. Nothing was ever off the table for me. I climbed trees, i had this sweet orange floral dress that I loved and wore it until it became a top. It was never questioned if I was 'allowed' to be dirty or play rough or anything like that, wearing a dress or skirt or anything else. Mum made a lot of our clothes and they were mostly gender neutral and equally unfashionable.
When I got older my dad showed me everything. It was just normal. Whatever he was doing he would call me over and show me exactly what he was doing, why, and how. Skinning an animal, chopping wood, or fixing something. I usually wouldn't DO it, but I would watch and learn. When I learned to drive he also taught me how to change a tyre, basic maintenance, and how to reverse a trailer. All of which i consider crucial information. I went into an extremely male dominated industry and it honestly never even occurred to me to be intimidated by that or to feel out of place.
One thing I think is most important to know - they've done studies about what gives a child good self esteem. Telling them they're great and special and wonderful is actually worse than doing nothing. What created good self esteem is teaching them actual skills, letting them experiment and learn and be self reliant. Then when your daughter comes out into the real world her self-esteem and confidence will be built on the solid knowledge that she CAN do things as well as anyone else.
I don't have kids of my own but one of my friends has a 12 year old girl and I'm blown away by this kids confidence. I think it's because her mother always let her do things for herself at an early age, e.g. at the age of 6 she learned how to haggle at garage sales (and got good deals too because people are caught off guard when a 6 year old is haggling expertly).
Build her up by making her resilient. Teach her things, and let her learn how capable she really is at a young age. By the time she's old enough to recognize the limitations other people are trying to put on her she'll have the confidence to blow right through them.