Why does it take 10 days?
Because it's not powered the whole way. You fire the engines briefly from earth to get to near-escape velocity, then coast to the moon, all the while slowing down because earth's gravity is pulling you back. It's like kicking a ball up a hill to get it into a dip over the top: you need to kick it hard enough to just get over the hill - i.e. when the moon's gravity is stronger at that point than earth's. If you went powered all the way you'd need a lot more fuel, and you'd also need even more fuel to slow you down when you got there.
