Unfortunately, this answer is going to be somewhat like the previous one. The challenges of sports and other action photography are twofold. First, by its very nature, sports photography tends to involve rapid motion - fast-moving players or cars or whatever. Second, usually there’s some distance between the action and the camera.

Solving the first problem requires lenses which can let in plenty of light, the use of flash or faster film or high ISO settings on a digital camera. Each of these solutions has drawbacks, however. Fast lenses are large, heavy and expensive. Fast film or high ISO settings result in higher grain or noise and thus lower picture quality. And flash may be inadequate to illuminate the subject effectively, particularly if the subject is some distance away.

Solving the second problem basically requires the use of long telephoto lenses. However, most affordable autofocus telephoto lenses are very slow - they don’t let in much light. So this amplifies the first problem.

Now obviously there are some cases where these two issues aren’t a massive problem. For example, perhaps you’re shooting a basketball game and you’re in the front row. Basketball courts are of a modest size and so you could probably do okay with flash (assuming you’re allowed to use flash - some places won’t let you as it can temporarily blind or distract the players) and you won’t need an incredibly long lens accordingly. Such a situation is a little less challenging than shooting hockey on a big, poorly lit, rink.

Nonetheless, pro photographers rely on fast lenses, and this is the primary stumbling block for amateur photographers on a budget. Fast telephoto lenses, especially fast telephoto zooms, are really expensive. And there’s not much you can do to work around that fact without a lot of compromises. To cover these points further:

Fast lens. Get the fastest (largest aperture) lens you can afford. A 70-200 2.8 lens is great for basketball, for example. A 75-300 4-5.6 is probably not, since even shooting wide open means you’ll have slow shutter speeds, which will result in unwanted motion blur.

Telephoto lens. You’ll need a long telephoto zoom unless you’re planning on shooting very close to your subjects. For example, you won’t need a long lens to shoot skateboarders in an urban setting, but you will if you’re covering a football game.

Cropping. You can always make up for a long lens to a certain extent by cropping the picture - trimming off the edges. The problem with this is that enlarging the picture also enlarges the grain in the case of film and lowers the apparent resolution in the case of digital.

Image stabilization. Useful for reducing blurring caused by camera motion, but of no value whatsoever in freezing subject motion.

Flash. Useful both for illuminating the subject and freezing subject motion. Not every venue permits flash usage, however.

Film/ISO speed. Fast film or high ISO settings are needed to keep shutter speeds to a minimum. Once again this involves tradeoffs with picture quality.

Camera with fast focus. A fast pro camera (such as the EOS 1 series) can lock focus surely and accurately and has minimum lag time when the shutter release is pressed. A consumer camera is not going to be as surefooted and decisive, and will make it harder to nail the perfect shot.

Fast lens motor. A Canon ring USM lens can autofocus rapidly, whereas a Canon AFD (arc form drive) lens cannot. A lens with a rapid motor frequently makes the difference between achieving a shot and getting nothing.

To summarize - if you plan on putting a 75-300 4-5.6 consumer lens onto your camera, don’t expect photos like those which grace sports magazines. This isn’t to say that you can’t get satisfactory photographs with such equipment, just that it’s challenging to do so. It takes a lot of skill, experience and luck to come up with consistently good results. And you should be operating on the expectation that you will face problems with blurring of the subject and general low sharpness and low contrast if you use an affordable consumer telephoto zoom lens.