"...how do the people that have known you for the longest time adapt to the fact that their perception of you is wrong?"
In order for those closest to me not to feel uneasy, I mustered all of my confidence and composure, and tried to show the continuity between what they know and what I'm telling them now. In other words, I emphasized that their perception of me hasn't been wrong, and that what I'm telling them makes sense with what they already know.
Unless you're transgender you'll never fully understand what it's like, so it was a difficult concept to explain, but I tried to make my explanations airtight ahead of time, and had plenty of analogies ready. As a result, the way they adapted was to first feel sad for me and a little upset that I didn't tell them sooner, but eventually they moved forward and now my relationships are closer than ever.
"So how do I tear down what has taken decades to build and replace it with something new?"
Before coming out I did the preliminary work and already was myself. The only things that might change now is that I'll look more like a female version of myself and I'll speak in a higher pitch. All of this takes time, so find a comfortable pace. Also, for everyone's sake, find joy in the process.