I will spout some ideas, please dont take any of them as set in stone rules. Try things out and take the things that are of use, discard any things that dont work for you.
When people think of a fast kick or punch they often think only of the limb speed (ie how long it takes a strike to connect after the instant it is launched). A fast kick or punch will probably have fast limb speed, but as well as this there are a number of other contributing types of speed.
Fast reflexes allow a strike to be launched as soon as possible after an opening is spotted. Reflexes can be trained by practicing reacting as fast as possible to a noise, signal, or to a partners movements. To help yourself react quickly, you need to be in a position where you can perform your technique with a minimum of unneccesary preparation movement. A relaxed stance with bent legs and evenly distributed body weight between your two feet means you can move in any direction without having to make a big preparation movement.
Limb speed is developed by loads of repetitions, I wish there was a shortcut but I think you just have to sweat it out to get faster. Relaxation is very important here, if your body is not extrememly relaxed then your muscles will compete and slow the strike down. Practicing things like yoga and tai chi can do wonders for your ability to relax. Also, a lot of people focus too much on pushing the strike out at the target and forget the second half of the movement, the return of the strike. If the strike returns as fast as possible it means that you are touching the target for a very short time, meaning that the impact is very sudden (and thus, hurts a whole lot more!) Practicing strikes going out slowly with a fast return can help with learning the co-ordination, it feels very strange to start with (and will probably leave you aching the next morning).
Another area that can make huge differences is giving your opponents less time to react. Part of this can be done by not telegraphing, and part by clever use of distancing. To avoid telegraphing, you need to practice launching strikes with absolutely zero preparation. Either find a big mirror or shiny window to watch your own techniques in, or find a good friend who can watch and let you know what you are doing to give away your intentions. As with fast reactions, the best way to not telegraph is to always be in a stance that allows you to launch techniques without preparing. To kick without preparing you need to get a feeling of lifting the foot to kick your opponent, and your body almost 'falling' towards them. You shouldnt have to squat down and leap at them, if your legs are bent sufficiently you can leap without bending any further. Front leg kicks can be launched without moving your supporting foot at all, just by lifting your front foot to kick the opponent and 'falling' into them. Distancing is a part of speed that is often forgotten. If you launch a kick from half a metre away it will take a lot less time to connect than if you have to move in from two metres and kick. If you can maintain the distance from your opponent that keeps them from attacking you but only just, then you have less distance to move when you want to attack them. And by throwing strikes that arent at your full reach, you can lead them to think that your range is smaller than it really is, so that when you throw a real attack you can reach them easily.
Sorry that each of these areas are so brief, if you want me to explain any of them more fully, email me at firstname.lastname@example.org.
I hope that you get even some small use from something that I said. Good luck for your tournament!