However, for some reason unknown to me, the visible matter in our solar system perfectly describes how the planets orbit the sun, how the moon orbits the earth, and how hard I hit the ground when I try to fly. So where is this dark matter, all this extra gravity? Shouldn't I hit the ground a lot harder than we can explain just based on the mass of our planet?
It's because dark matter only interacts gravitationally. See, normal matter clumps up into planets and stars because it sticks to other particles, and loses energy from collisions, causing it to collapse over time into locally dense spheres (planets, stars, black holes, etc.). But dark matter doesn't: it just passes through itself (mostly: it may interact through the weak force, but only very very very rarely if so, not enough to clump up). That means it doesn't form local regions of high density. On the other hand, an object immersed in a more or less uniform sea of matter (of any kind) won't notice any gravitational effects, because it's being pulled in all directions equally (for example: you'd be weightless at the center of the Earth. Dead from the pressure/heat/lack of air, but weightless). So, we can float through a sea, even a fairly dense one, of dark matter and notice nothing at all. Now, there is an non-uniformity in this dark matter "sea": there is more on the side of us towards the center of the galaxy than there is on the other side, but that pulls the entire solar system uniformly, accelerating it in it's galactic orbit, and that effect we do in fact see.