Best Nikon D5100 lens for kid sports?

I just got a D5100 kit and i need a good lens to take pictures outdoors and sometimes indoors. I take pictures of youth sports such as soccer, baseball, basketball and hockey.what's a good fast VR lens i can buy? I would like something under $400.under $300 would be great.

i used a canon rebel xsi w/ 75-300mm f/4-5.6 III for a hockey game. The lighting was LOW and the auto focus wasn't that great. I missed a lot of good shots cuz the lens couldnt focus fast enough. Also some pictures were blury because of the low lighting

so DO NOT want to go through this again so.what's a good lens?

Sorry but there isn't much fast lenses in te price range. By fast youprobably mean F2.8. The only sport lens 'new' I can recommend is the 70-300mm VR. This lens is pretty good. It has VR, ED glass (extra low dispersion) glass. It also has IF, internal focus. This means it focuses internally so the end barrel doesn't rotate like on
your 18-55. The only main problem is it is F4.5-5.6.It is only in good use with good light. When it gets darker you may have to push ISO.
To avoid this problem you could buy a Sigma 70-200 F2.8. This will stay at F2.8 through the zoom range. There's one for £1k wich is out of your budget. There's a second version for £562 new or £545 used. It has APO which is basically Sigma's version of ED. It has a 1:2 macro function do it can ficus closely. It is an excellent lens. I used one at silverstone and it, s focusing speed is quick. It hasn't got any form of VR but it doesn't matter. It is a 70-200 so it is too heavy to hand hold. I recommend using a monopod so there us no need for VR.

So you have a choice.
Save some cash and get NIKON 70-300mm VR ED IF F4.5-5.6 G. Good lens but doesn't let in as much light.

Spend more on Sigma 70-200mm F2.8 APO EX DG Macro. Faster aperture, faster focusing.

You will be much happier in the long run with the Sigma because it is better glass.

Hope this helps. Got any questions just ask me.
James

As James said, your're not going to get much lens for that money. If you can't afford or don't want to spend more, you could buy flash gun, this will have a much better range than your built in flash and can be used to 'help out' light levels with closer shots on dull days, and will really help with indoor situations.
For baseball shots you could use a tripod and manually pre-focus on the batter. It won't improve your shutter speeds but will keep the camera still and a little movement can help in sports shots.
I know this isn't really answering your question but it might help a little.