reaction drills.
"out of the gate" speed is all about reaction. if you react quicker than the next guy you are going to win.
seeing as out of the gate stuff is only 1 second or less what you need to focus on is reaction type drills.
here are some good reaction drills.
1)lie flat on stomach and have a partner blow a whistle while u sprint a certain distance. usually 10-15 meters.
2) lie flat on stomach and have partner a certain distance from you drop a tennis ball to the ground . you have to catch the ball before it bounces a second time. start at a short distance and increase it as you get better.
3) work on your vertical. stand beneath a bball goal and explode to reach your hand as high above the rim as you can. 3 sets of 10 is good for one workout as you start. add more sets as your vertical will increase pretty fast when first starting. look online for tons of vertical jump exercises.
i guarantee you that doing this drill alone will help more than any weight lifting routine for your quickness and explosiveness.
but in the weight room speed squats work great. make sure the squat is a good 3-5seconds at the bottom and explode up.
also full cleans are very good for explosion.
what sport are you trying to develop this quickness for?