Jen, welcome to the Hog.
Don't know if you have looked at any equipment yet, but you will see that $1200+ just doesn't go very far. The two most expensive pursuits are wildlife and sports.
Maybe you can get your kids to switch to the chess club!
I think your major consideration is going to be the lens. You need long and fast to get rid of that fence.
But FYI, a good lens, if you use it for a year or more will sell used for close to what you payed for it, then a more appropriate night lens can be purchased when the time comes. Lenses are NOT like new cars, that when you drive it off the showroom floor, thay are already worthless.
So, a lens such as an f4 that works great in daylight but not so well for night games, can be replaced at a later date with a very minimal depreciation.
If your shots tend to be static, even inexpensive bodies should do a pretty good job. If you're trying to get base runners sliding, or the second baseman jumping over a slider and throwing to first, it will be much more critical that you have a sports body. Sports bodies have special focusing systems that track the fast action better than regular bodies but tend to cost more.
For a less expensive package, I recommend you look at a Canon 7D and a non IS 70-200, F4 zoom. Good for daylight. Refurbished and on loyalty from Canon the body is about $1000. The lens refubed is about $600.
A note, though the Sigma lenses being recommended are much cheaper than the comparable Canon model, they also tend to be much heavier. Meaning you can loose the versatality of hand holding a lens and be forced to use a tripod.
I believe, though I'm not so familiar with it, that tha Canon 50d body is a pretty good sports body and much cheaper, but not sure of its megapixels. Maybe someone can chime in on the 50d.
FYI, sports bodies focus better and have very high frames per second abilities.
Hope this might have helped.
Jen, again welcome to the Hog.
SS

Your budget is approximately $1,200. Some suggestions.

First, go to a good camera store to test as many cameras as you can. Even Best buy is better than nothing. How the camera feels in you hands will be an important factor.

Second, do as much reading/research as possible beforehand so that you will have some idea as to what the salesperson is talking about.

Third, an inexpensive body using a great lens will take much better photos that an expensive body with an average lens. These days the glass is everything inasmuch as entry level bodies can take great photos with the right lens.

Fourth, while most here use Canon or Nikon, be willing to look at other makes. The new Olympus is a great camera, Sony and Pentax also make some very good cameras.

Fifth, an f4 lens is the slowest you want to buy. For tack sharp action the f2.8 is by far the best way to go. As for me would get the 70-200 f2.8 to start with. Even a used one if necessary.

Lastly, While $1200 seems like a lot, to get outstanding action photos in all kinds of light you may have to up the budget. Good glass is not cheap. And I can almost guarantee that if you start out with lesser quality lenses you will soon be wanting to or even upgrading the glass. While spending more at first may hurt you will save a lot in the long run.

Best of luck and be sure to have fun.

NO!

The numbers you repeated are the focal length of the lenses.
The larger the number, the more it magnifies.

The 'speed' is specified by the f number and it's reversed from what you would think. A smaller number lets more light in than a higher number.
Most people shooting action sports buy lenses with f number of 2.8 or smaller. And the smaller the number the more they cost.

I rented a Nikon D7100 with a 70-300 f 2.8 for shooting High School football under the lights. As with any higher level camera, it takes a while to get used to it and understand how to make it do what you want, but the images are pretty good.

As someone else mentioned, I would suggest you consider a Nikon D7000. It's an older version of the D7100.
Adorama (a very reputable dealer) has a kit with a 55-200mm lens for just over $1000, with a $200 rebate, putting it right in your range.
http://www.adorama.com/INKD7000L4.html

Whatever you end up purchasing, I *STRONGLY* recommend that you get some hands-on training on how to use your new camera. And how to deal with the images it makes.

She would be represented by a plaintiff lawyer who will accept as payment a contingency (a share of the amount.) Expenses will also be paid out of the verdict. If defense wins, neither the plaintiff
nor lawyer is paid.

In this hypothetical situation, the civil defense lawyer will be paid by the hour plus expenses.

Economics dictate a plaintiff lawyer is unlikely to accept a case with little chance of winning. But there are many exceptions to this general rule.

The plaintiff lawyer is required to prove damages. My guess is the harp player may be entitled to some damages, nominal, if the photos are published without her permission. If the web site "sells" the photos, then actual damages may occur as well as punitive damages, all owed by the photographer.

The question as I saw it is digital, not analog. Should the photographer publish the photo without a release? Digital answer is "no."