First, recognize that the camera has two entirely different focusing and shooting behaviors.
The default behavior is intended for non-moving subjects. This mode (which the camera calls "One Shot" mode) puts priority on focusing once and then taking the shot. The camera will focus until it achieves a focus lock. As soon as it does achieve a focus lock, it STOPS trying to focus and if you completely press the shutter button, it'll take the photo. The "problem" with using this mode for cycling is that once the camera locks focus and stops focusing, your subject is still moving. They are now no longer at the same focused distance where the camera achieved a focus lock and the camera is not trying to keep them in focus. The result is an out of focus shot.
The other behavior is "AI Servo". This mode is intended for sports or any shot where the subject is moving. The camera never stops focusing. Once it achieves a focus lock it does not stop focusing, it continuously re-checks focus.
There's one nuance of "AI Servo" mode. It also changes the behavior of the relationship between focus and shutter release. In "One Shot" mode the priority is on focus. If you simply mash the shutter button down completely, the camera won't actually take the shot instantly... it'll perform a focus, and the moment it gets at least one AF point to lock focus THEN it'll take the shot. This behavior is called "focus priority". But in "AI Servo" mode the camera changes to "release priority". In that mode the camera WILL shoot as soon as the shutter button is completely pressed whether anything was actually in focus or not. As the sports photographer, it's up to you to decide when to shoot. So you do want to confirm that your satisfied with focus before you completely press the button as the camera will no longer 2nd guess your decision to shoot -- it'll just follow orders.
There is a third "mode" which is really a pseudo mode. It's called "AI Focus". That mode is really a mode that causes the camera to auto-decide whether it shoot use "one shot" behavior vs "AI Servo" behavior on a shot-by-shot basis. The trouble is there's a slight delay while the camera evaluates whether the subject is moving or not. So if you KNOW your subjects are moving, just use "AI Servo" mode for better performance. If you KNOW your subject isn't moving, then just use "One Shot" mode.
BTW, the logical next question is... why not just leave the camera in "AI Servo" mode all the time -- since it sounds like that's a better focusing policy. Turns out "AI Servo" mode isn't just ready for a moving subject... it assumes ALL subjects must be moving. The slightest nuance in any way will cause the camera to re-focus and it can result in an out-of-focus shot when used on non-moving subjects. Canon recommends you always use the right mode for the job and don't just leave it in one mode for everything. If you really want to be lazy... use "AI Focus" (the "auto-decide" mode) but just know that this will cause a slight performance delay while the camera evaluates which mode to use.
Now that you've got the camera in the right focus mode (very important), the next question is... how fast is the focusing motor on your lens? The entry level lenses such as the 18-55mm kit lens, the 55-250mm zoom, etc. don't have very fast focusing motors. For sports, you want a lens with snappy focusing motors. Look for lenses with the "USM" tag in the name (USM = ultrasonic motor). The USM focusing motors are much faster (there are different types of USM focusing motors and some lenses are quicker than others.) Using a lens with a slower focusing motor means the camera may always lag just enough to prevent you from getting sharp focus if the focus distance is changing.
The advice on "panning" is great advice, but that's a slightly advanced technique. You should be able to capture sharp non-panning shots first. If you can do that, then the next step up to evolve the look of your images is to learn to "pan". This means you deliberately reduce the shutter speed below what is necessary to freeze action, but you pan the camera to carefully "track" your subject. When you snap the photo, the subject will be sharp (because you were tracking them carefully) but the background will be blurred. The photograph mentally "implies" motion to anyone who looks at it.
Here's an example of a "panning" shot.
Joyride by
Tim Campbell1, on Flickr
Notice that while the car and it's occupants are sharp, the spokes are blurred as is the foreground and background. This was taken at 1/50th sec.
Panning takes practice. Start at a faster shutter speed. Usually 1/500th will "freeze" almost any action shot (unless the action is VERY fast). Try dropping to just 1/250th and take several shots. Then try dropping the speed down to 1/125th and take a few more. If you are still getting good results, go for 1/60th. With practice you'll be able to get the speed down. Expect lots of bad shots as you learn the technique. The camera needs to be well supported and braced and you need a nice smooth steady movement as you track your subject. You'll probably notice that even at 1/125th you have the nice effect of background motion and don't necessarily need a particularly slow shutter. The optimal speed to imply the motion depends on how fast the subject was moving. The car above wasn't moving particularly fast when I took this -- so I needed to get the shutter down to 1/50 or 1/60th to get the effect (at 1/125th you wouldn't have noticed much). But if the cyclists are moving at a fairly high rate of speed... then 1/125th may be ideal and you may never need to drop to 1/60th.