I just got my HC-06 module and tried installing it. What I've found interesting is that it won't work proper if it's connected to the dedicated Serial lines (A0 and A1 in Arduino Leonardo). But, if I make a Software Serial port it works fine (actually great). Probably it has something to do with the Serial port being used by the USB. But, I am not sure why this is and how can I make it work via those lines. I'm using Arduino Leonardo, so only 1 Serial available.
Thanks!