Hydraulic disk brakes starts braking after pulling the half of the lever, how can I solve this problem with the brakes?
I think it depends on pad adjustment and most hydraulics have some kind of auto adjustment, also following pads' wearing - small "springs" between the pads return them to the initial position. Cable driven disc brakes can be adjusted at the caliper for tighter fit. 
You'll probably never be able to start braking immediately. It needs some play, otherwise it would always touch the disks.
Though wiht hydraulics you should also check if you have enough oil/brake fluid in the lines, without air in it. That can also cause a loose, weak braking. It's probably done best by a professional.