Like a lot of things associated with photography, a lot depends on how much you are willing and/or able to spend to get that "next big shot." A lot also depends upon the conditions under which you'll be trying to get that shot. There's a big difference between what you need to shoot moving horses under bright sunlight and shooting the same activities under artificial lighting at night or at an indoor arena.
At the top of the heap would be a 300mm f/2.8 or even a 400mm f/2.8, 400mm f/4, or an even longer focal length Super Telephoto prime lens. That's what you see the top pros shooting outdoor sports using. The 300mm and 400mm f/2.8 offerings from Nikon and Canon are some of the best performing lenses in the world. But most Super Telephoto lenses are certainly not cheap!
In exchange for a lower price than the Super Telephoto prime lenses, you need to decide what you are willing to give up for that lower price. (Generally a telephoto lens with a focal length 300mm or longer is considered a Super Telephoto, but many people use the term in a more restricted way to indicate wide aperture primes that are 300mm or longer. Zoom lenses that end at 300mm are usually considered "telephoto", rather than "Super Telephoto.") Which can you do without and still get "the shot?" Large aperture? Absolute image quality? Longer focal length? The Nikon name plate on the lens?
Sigma's latest in their series of 120-300mm f/2.8 zoom lenses is highly regarded by some as a good compromise between the absolute image quality of the Super Telephoto prime lenses and the flexibility of the variable aperture zooms. It's a bit pricey but is still the cheapest way into the 300mm @ f/2.8 club.
If you are shooting in bright daylight an option a lot of shooters like is one of the recent 150-600mm f/5-6.3 offerings from Sigma or Tamron. At the long end of their focal length range they're not quite as good as the high end primes, but they're still pretty good. They're also nowhere near as expensive. And while they are very usable in bright daylight, they're not so usable for shooting action under dimmer artificial lighting.
There are many cheaper options that have focal lengths in the range of 70-300mm and variable apertures in the f/4-5.6 range. One thing to remember with variable aperture lenses is that the longest focal length also has the narrowest maximum aperture. At 300mm the 150-600mm lenses are at f/5.6 compared to the 70-300mm lenses that are 1/3 stop narrower at f/6.3.
Lenses with even wider focal length ranges, such a 55-300mm or even 28-300mm or 18-300mm sacrifice image quality, particularly at the longest focal lengths in exchange for their versatility as an "all-in-one" lens. Although it concerns specific lenses for the Canon EF mount rather than the Nikon F mount, the issues are the same at Why prefer the 18-55mm and 55-250mm lenses vs 18-200mm?
If you are shooting in lower light then you probably need the wider aperture of an f/2.8 lens or even f/4 constant aperture zoom more than you need the focal length of a zoom that is f/6.3 at the 300mm end. f/2.8 and wider lenses are called "fast glass" because using the wider aperture allows a faster shutter speed for the same amount of light and the same ISO setting. At f/2.8 you can use a shutter speed five times shorter than at f/6.3 for the same ISO. Even at f/4 you can use a shutter time 2.5X shorter than at f/6.3. Nikon offers several stabilized and non-stabilized versions of 70-200 f/2.8 and f/4 zooms. There are also some very good older 80-200mm f/2.8 Nikon zoom lenses on the used market.
(The 80-200mm lenses are capable of autofocus, but only with a Nikon body that has an autofocus motor built into the body. Your D3000 requires lenses with AF motors in the lens in order to autofocus.)
Vibration Reduction/Image Stabilization isn't a key consideration when shooting sports/action such as leaping horses. Image stabilization can be helpful to minimize the effects of camera shake when shooting handheld at slower shutter speeds. By the time they reach 300mm and beyond most pros are using monopods to help support the weight of the lens. They do it as much from a consideration of endurance during a several hours long sporting event or shooting session but it also helps with reducing camera movement. Image stabilization does nothing to reduce the effects of subject motion on a captured image. It can sometimes come in handy when doing a panning shot to help keep the subject sharp as the background blurs behind the subject you are following with your lens as the shutter is open. It can also make the scene through the viewfinder more stable at longer focal lengths.
What am I looking for in a lens to get the pictures I want? AF, VR, G.....what?
For help in understanding what all of the numbers and letters in a lens' name mean, please see: What do all those cryptic number and letter codes in a lens name mean?