There is speed, there is quickness and there is reaction time.
You need to train all 3 ( and any other you can think off).
Richard is quite correct when he speaks of relaxation and correct technique (minimum motion), they are crucial.
There are many drills but the key elements is how you then apply the attributes that you have built into fighting.
Of course the answer is: sparring.
One thing we MUST be aware of and that is: Follow-through
Many times when we train speed we end up "flicking" our techniques and lose impact force.
We must be very thorough in our development of correct follow-through (penetration) as well and that is where hard/full contact training comes in.
Psalms 144:1
Praise be my Lord my Rock,
He trains my hands for war, my fingers for battle !