I always receive some very bad sidekicks when I try to go offensively while in kickboxing. Mainly because I try to make a jab with my right hand but the kick comes too fast (or I'm too slow) I have countered this a bit by putting my left hand on the side while making the jab to protect myself. It is sometimes effective but lowers my defense on the left side of my face.
So I want to address the problem at its root: footwork and speed. I will not ask which is the most effective way of improving those because it is too subjective. I am asking about an effective way to improve it and that is easily accessible (i.e. no need of special facilities or equipment)
For instance I have heard that to improve footwork, dancing is effective.
Footwork is not just about moving in the right directions, it's also about getting there quickly and being in balance as you do it. Footwork will be no good to you if you are a lumbering elephant with no balance or dexterity.
A couple of ways to get lighter on your feet are:
skipping. While used extensively in boxing for fitness, it also teaches you to stay light on the balls of your feet. Once you can easily do one pass of the rope for one jump, double it up and do two passes of the rope for every jump. Before you know it you will be bouncing around everywhere.
find some creaky stairs and run up them - quietly. This teaches you how to place your feet and move your body weight through a range of motion without pausing or sticking in any one spot
take a karate stance such as zenkutsu dachi and do it up and down a beach. Doing long lines of this in the sand helps you learn to glide with no up/down bobbing as you transition between left and right versions of the stance. The sand will show you whether you are digging in or bouncing around in any particular spot. Your feet should leave nice grooves of equal depth all the way down the beach. While this is usually a slow exercise, it is training your mind and body in the correct way to move and is just as applicable when moving fast.
practice a snap front kick. Pay attention to the end of the kick - don't just let your foot drop down back into place, instead place it back in its original spot in a nice controlled manner. Pick your foot up, snap it out, place it back down. This helps train you to pick your feet up fast, maintain your balance, and best of all the snap front kick can be used as a closing-the-gap technique. Keep practicing the kick even when you are tired as that will be when your technique starts to get sloppy and you go back to your old habits.
practice multiple kicks. Doing a single kick from a fixed position is all very good, but doing combinations of kicks one after the other forces you to pay attention to your balance and will also improve your footwork.