What equipment should i use for taking good action shots at night on a nikon d5100?

What equipment should i use for taking good action shots at night on a nikon d5100?

Basically, a tripod. If you want more, a remote shutter release.

Action shots at night means you need a fast lens to capture as much of the available light whilst taking shots of a moving subject. That usually means a lens of f/2.8 or better, like the excellent Nikon 80-200 or 70-200 VR zoom lenses (btw, make sure it's an AF-S or G lens, as AF-D lenses won't autofocus on a D5100). However, as a rule, f/2.8 lenses are quite expensive. You could try using a f/1.8 50 mm prime lens, which is much cheaper (and still very good).

Alternatively, if you need more zoom, you could use something like one the 18-200, 55-200 or 18-55 mm lenses with vibration reduction (VR). These are not as fast as a f/2.8, but the VR-feature will help reduce motion blur.