How to adjust Shimano hydraulic brakes contact point? The title might be wrong, I am not familiar with the terminology here.

I just got a BR-M445L front hydraulic brake. Installed it - this stuff is MUCH easier to install than a cable disc brake.

Now my problem is that in order to apply maximum braking force, the lever has to be pulled all the way to the handlebars. 

I discovered the reach adjustment screw on the lever. With this screw, I moved the "at rest" position of the lever further from the handlebars. Thus, the situation is somewhat better now.

However, what I would want to do, is leave the lever not too far from the handlebar, and remove the "free travel" - the part of the lever's path, during which the pads do not contact the disc at all.

In other words, I would like to have 2-3 cm of travel of the lever, instead of 5-6.

There are a couple of screws on the calliper, but I am scared to touch them, before knowing what I am doing. One is on the entrance of the hose, similar to a cable brake adjustment screw. The other is at the base of a small opening, covered with a cap.
The brakes should adjust themselves automatically. Try pumping the lever all the way a couple of times. If that doesn't help, or the contact feels squishy, the brakes probably need bleeding. Either do that yourself or take the bike to your LBS.