This is a classic question that I remember seeing when I was a high school student: "How long would it take to fall through a tunnel through the Earth?" My Physics teacher solved the question for us and I'll have to confess only understanding the solution for about 30 minutes. Today I went back and re-worked the question myself. I've presented the solution in two parts.
First, what is it like to fall straight through the earth:
Second, what if the tunnel doesn't go through the Earth's center? What if you want to connect NY to LA for example?