RavenHorse, What is your budget for camera and lenses?
A couple things to remember. One, its not as much the camera that makes the images as it is the photographer. You can put a $5000 camera with a $8000 lens in someones hands and still not get a better photo than if they were using a $250 point and shoot camera.
Knowing what makes a photograph better and how to
control the camera to get those results is what makes the difference.
Two, knowing/learning how to control your cameras shutter speed, aperture, ISO, depth of focus, and their relationships are technical things to learn. Be prepaired to get a little technical.
Lastly, DSLR's and lenses can vary greatly in price. A starter kit can start out in the $550 range. This will give you a camera that will capture the photos you want just fine, but the lens that comes with it would need upgraded to achieve the photos you describe. This leads to buying a telephoto or telephoto zoom lens. For tele/zoom lenses you have several options, cheap, mid priced, high price range. Lets say you decide you want something that will give you 200mm(about 4X zoom) or up to 300mm(abotu 6X zoom). You could go with a 70-300mm F/4.5-5.6 lens for $160 or one slightly better for $586. These two lenses are the cheap options. Next step up would be a zoom lens the 70-200mm F/2.8 for $2396 or a 300mm F4 for $1300, this would be the Mid-priced range. You could go up and get a 200mm F2.0 for $5819 or a 300mm F2.8 for $5800, this would be into the upper price range. Lastly, maybe you figure that 200mm and 300mm lenses just aren't enough and you need a super telephoto lens to be able to shoot all the way across the horse ring. You options now are a 400mm F2.8 for $9000, a 500mm F4 for $5400, a 600mm F4 for $9800.
Now do you need to spend $10,000 on camera gear to get great images, NO! At the same time if you get the bottom of the price range lenses you are going to have a MUCH harder time getting those great images even when you know what you are doing with the camera.
Hope this gives you a little insight into what you are in for. Good luck and read up as much as you can.
Not a big horse photographer, but here are a couple shots you may enjoy.
A friends Colt
Preakness 2008
Jason
