There's the advice-to-a-noob response of, "Do the tubes use Presta valves? If so, are you unscrewing them first to open them?" But, you have a floor pump already, and you know how to use it, so that's out.

Then, whether you have the lever flipped the right way or not shouldn't be causing this problem, because if it's flipped the wrong way the air will be leaking out the sides instead of getting stuck between the pump and the valve.

The one thing I can think of is to make sure the pump is really jammed onto the valve stem. My floor pump looks like it uses a similar kind of head, and it won't pump at all if it's not on the valve far enough. It's something about the check valve inside.