For technical aspects, the key is two-fold:
1) Use a large enough aperture to let enough light in. In my experience, this is usually f/2.8 or larger, unless your field has very bright lighting.
2) Use a camera that has a relatively noise-free sensor at ISO 800-1600. This inherently rules out virtually all point-and shoot cameras and a good number of digital SLRs.
I've had good success with Canon EOS 20D and 40D bodies, and with the f/2.8L zooms.
To capture interesting photos, there are three important points to remember, but they are vastly different from the former two:
1) Ensure your shot is a verb, not a noun. Don't photograph a robot. Photograph a robot doing something. Don't photograph a person. Photograph a person doing something. And, try to show that action in your shot. Sometimes blur is effective for doing so, but it's not always necessary.
2) Try to get as close to the field of view of the human eye as possible, without a) compromising your composition and b) being too far away from your subject. Getting closer with a telephoto lens is no substitute for using a wide angle lens and physically getting closer to the action. That's what makes a viewer feel like they are "there", when viewing a photo.
3) Capture a vantage point that is atypical to the average bystander. Usually this means having your camera higher than 6 feet or lower than 2 feet, when shooting. You'll be amazed at how much interest this can generate, just because we are unaccustomed to this perspective. This is especially true when photographing small people (like children) or animals -- do so from their eye level.
The human eye has a field of view close to 180 degrees, when including peripheral vision. A telephoto lens will have a field of view in double or single digit degrees. That's only a fraction of what you would see if you were at an event in person, so it ends up looking like you're watching the event through a window.
On a Canon EOS 20D/30D/40D, your best option is the EF 16-35mm f/2.8L II USM or EF-S 17-55mm f/2.8 IS USM. On a Canon EOS 5D/1Ds/1D, your best option is the EF 16-35mm f/2.8L II USM, because of the flexibility of a zoom, or the EF 24mm f/1.4L USM, because of the much wider aperture.
Yes, those are expensive lenses, but the only way you can get around that is by introducing more light. That usually means flash. Bouncing isn't an option in an open arena, IMO, so that leaves a diffuser. However, even with a diffuser, you are now changing the actual scene and the background will almost always be darker than it would without the flash.
If you can't afford a new, fast lens, try a used one. Even the original Digital Rebel (6.3 Megapixel) body with an f/1.4 lens will yield better results than the 1D Mark III with a slow lens.
My most recent examples are here:
http://robotphotos.org/v/first/fingerlakes/2008
The absolutely cheapest setup I can recommend that will yield better options than your point-and-shoot is the original Digital Rebel with the EF 50mm f/1.8 II. However, you should be aware that the 50mm focal length on the Digital Rebel will have a field of view similar to an 80mm lens, which is considered telephoto. So, it's not going to have a wide field of view. The EF 35mm f/2 would be wider, and slightly more expensive, and a little bit slower. The EF 24mm f/1.4 would be wider yet, and much faster, but not as wide as the EF 16-35mm f/2.8L USM (or the II version).
The EF-S 10-22mm f/3.5-4.5 USM is just not an option, because at f/3.5 it's marginally too slow, and at f/4.5 it's way too slow.