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.

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.

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

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.

Jenni,

Welcome to UHH. Nobody has yet answered your initial question. So, the best camera for sports shooting from Nikon is the D4 and the best from Canon is the 1Dx. The both shoot at a very high frame rate and at a very high ISO, both of which are critical to shooting sports. The best lens for sport shooting is less certain. For me it would be the 70-200 f2.8. Both Canon and Nikon make excellent ones. The argument for buying one or the other of these cameras and lens is that either will fit your needs for several years and you will not end up buying and selling several cameras and lenses only to end up where you could have been from the beginning.

I understand that it is daunting to think about spending that much money and being wrong, so, I suggest buying used or refurbished. A very good Nikon for fast frame rate and pretty good high ISO is the D3 model. I am sure that Canon has comparable models, but I am not sufficiently familiar with them to offer advice. Others here are, though. My last suggestion is to make dPreview. com you favorite web site, next to UHH, of course. You will be able to compare size, weight and picture quality of all recent and many older cameras, so you will not have to guess about what the various camera models can do. You should also google "sports shooters"' look at the pictures and see what equipment was used.

Good luck in your search, keep asking us questions and let us know what you decide on and how it works out.

Everybody has their own opinion. The only opinion which counts is yours. You have to do your own research. What is recommended by someone may not fit your needs.

For now put aside your budget. From a point and shoot camera (P&S), start reviewing bridge cameras with an attached zoom lens. Review online then go to brick and mortar (photo) store to hand hold cameras for your comfort level and your budget.

No matter what you purchase, it won't be your last camera because you will grow. Keep your P&S as a backup camera

As far as shooting through a fence?
1. try to avoid it or
2. get up as close to the fence as you can (lens hood on the fence ~ see note below) or
3. get into post processing (PP).

Note: a bridge camera may not have a lens hood, so if possible attached a UV or clear filter on lens. Hoods and filters are for protecting the lens from damage.

You have to do your own research. After you find two, three or four cameras you like, go on line and use sites which will compare cameras.

No matter which camera you choose, you can't go wrong. Master that camera, then upgrade to another camera, keeping the previous camera as a backup.

Keep in mind that you're use to have a light weight P&S which fits in your pocket or purse. With a bridge or a DSLR you may not have that convenience. Have fun.

Everybody has their own opinion. The only opinion which counts is yours. You have to do your own research. What is recommended by someone may not fit your needs.

For now put aside your budget. From a point and shoot camera (P&S), start reviewing bridge cameras with an attached zoom lens. Review online then go to brick and mortar (photo) store to hand hold cameras for your comfort level and your budget.

No matter what you purchase, it won't be your last camera because you will grow. Keep your P&S as a backup camera

As far as shooting through a fence?
1. try to avoid it or
2. get up as close to the fence as you can (lens hood on the fence ~ see note below) or
3. get into post processing (PP).

Note: a bridge camera may not have a lens hood, so if possible attached a UV or clear filter on lens. Hoods and filters are for protecting the lens from damage.

You have to do your own research. After you find two, three or four cameras you like, go on line and use sites which will compare cameras.

No matter which camera you choose, you can't go wrong. Master that camera, then upgrade to another camera, keeping the previous camera as a backup.

Keep in mind that you're use to have a light weight P&S which fits in your pocket or purse. With a bridge or a DSLR you may not have that convenience. Have fun.

Hi Jenni,because you budget is 1200 dollars and to satisfy you needs as of right now I would suggest a used or refurbished Nikon D7000 and a Tamaron 18-270 lens. With the crop factor of the D7000 the 270mm will be equivalent to 410mm which will nicely get you going. You can progress into a faster 2.8 lens when the time requires it as they are very expensive. I sure you will be happy with what I suggested as you will get a wide angle and good telephoto along with nice clear photos and within your budget.