Lens of Nikon D5100 DSLR?

I have a D5100 and it does well for me but I only have an 18-55mm lens. I'm still fairly new to this. I'm looking to buy lens or equipment that can focus better when taking action pics with no flash (for example sports). Also something with a quicker shutter speed. And I need something for video also, that will autofocus depending on if i zoom in on something and i'm using the camera on a tripod to record myself. What would you recommend?

Good lenses are expensive. The 70-200 f2.8 would be a great lens. Your camera controls your shutter speed. All you gain by having a faster lens is a larger aperture so that you can use a higher shutter speed. If you have a 200mm f5.6 and can only shoot at 1/250 because of the light, an f2.8 will allow you to shoot 2 stops faster (1/1000) under the same light. Also, you do not want to use autofocus when shooting video. Learn how to manually focus or your video will not look nice.

Depends what sports you are shooting…

and from where in the audience…

different lenses can do different things…

and there are many across the price ranges…

so your budget would dictate your options…

70-300mm lens would work for most sports where you aren't very close to the action…

but the f/2.8 version would probably be outside your budget…

55-200mm f/4-5.6 vr shuld be good enough. Use Iso 200-400 outdoor day sports and iso 800-3200 for night/indoor sports.