I have noticed that a lot of beginning motocross riders often get pretty confused about how to shift gears properly. Everyone seems to have a different opinion and it can be quite difficult to find instructional materials that actually cover this important aspect of motocross technique.
There are two ways that I personally like to shift up gears on a motocross bike and I'm going to attempt to explain them in this short article.
For the first way you do not have to use the clutch. Basically all you need to do is back off the throttle, shift up and then get straight back on the throttle as quick as you can. It's OK to not use the clutch in this situation as the drive is momentarily disengaged.
This way is fine but since you have to back off the throttle, you will lose a bit of momentum and you can not afford to do that in a race.
There is a much more effective way of upshifting that will actually allow you to go much faster and you will not lose any speed.
All you need to do to is get the throttle pinned wide open, pull in the clutch just a tiny bit so that the engine is just about disengaged, shift up then let out the clutch. Do this as quickly as you can and make it one fluid motion. This will allow you to keep the revs high so that when you let the clutch out, the power is there right away. You will not have to wait for the power to dial in again as in the previous method.
Now, a lot of people think you have to use the clutch when downshifting but that is not the case at all. When you shift down, there is not anywhere near as much of a strain on the engine so it's not really necessary. In fact, by not using the clutch you are able to use the engines natural resistance to help you slow down (this is called engine braking). Ideally, the only time you should be shifting down is when you are on the brakes. Being on the brakes means that there is even less strain placed on the engine and the combined effect will allow you to slow down a lot more efficiently.
I hope this will help to clear up any confusion you might have about shifting properly on a motocross bike.