The home button problem is fairly common and it is usually due to gunk/corrosion hampering the underlying connections.

There's an easy, tool-free fix: Liberally spray the home button with some Deoxit [which can be found at a RadioShack or on Amazon]. Then press the home button several times to ensure that the solution fully saturates the underlying connection. Wipe off the excess and let your phone sit for a while. Your home button should work wonderfully after that.