Update: With the help of the smartest guys in the world (the community here on Scripting News), it works and I now have two inflated tires on my bike.
I've been feeling kind of sick, but I must be feeling better cause I want to get my bike working!
After a winter of inactivity, both tires are flat. So I bought a hand pump. Here's the howto that came with it.
So I tried clamping the thing onto the valve first one way then the other (they don't make it clear which way is open and which is closed).
But either way I just get a brick wall. The pump won't go down. The tire remains flat.
I did manage to completely empty the rear tire by fiddling with the valve. Absolutely every last bit of air is gone.
Any clear, slow, step by step advice would be much appreciated.