Nikon D5100 bad sports photos INDOOR?

Recently got this camera (i have the kit lens and a 55-200, if that solves anything)
And i have a massive massive interest in horses, that's literally what the card will be full of, and i'm not impressed with any of the photos i've taken.
This is a fast moving sport/hobby in my mind so i can't be standing around changing settings after every photo i've taken, i'll be missing the shots/

This is indoor, on sports mode (tried the pets mode, was worse) and the 'indoor' mode was laggy with a flash

It focuses perfectly on everything around it, but usually with a sports mode there's a moving focus green thing, which is usually there on other cameras

I tried changing the shutter speed to capture it quicker but the photo was pitch black.

This was with flash, because i got fustraited.

It may seem i have no idea what i'm on about but i'm new to this, and i just want to take some decent photos after paying So much for it

Please don't just say "practice" because thats such a wide word meaning "turning the camera on and off properly"

I want to know what i'm doing wrong! Thanks!

Added (1). Thank you very much everybody.
Are there any lenses under £550 instead of a £3000 lens to help the shutter speeds?

You need a faster lens. The 55-200mm VR is a slow lens.

If you're using the built-in flash, there's no way it'll carry far enough even with your ISO high.

Most pros use f2.8 lenses in this situation because indoor lighting is pretty useless for photography, so with your existing lenses, you already have a mountain to climb.

The sports mode is the right one, but you'll have to use an ISO of at least 1600, probably higher, to keep your shutter speed high enough - then use software to try and mitigate the noise - I'd also recommend finding something to lean on.

The problem is NOT the camera, but the speed of the lens you are using

Here is a shot, taken indoors using a much older camera and it still produces relatively noise free images at 3200 ISO

It was taken using an old Nikon D300.

As you can see, your D5100 has a much higher performing sensor so should produce much better images than this D300 does.

http://www.dxomark.com/...nd2)/Nikon

So, you see, you need a faster lens and hone up on your shooting techniques

The shot was made with the white balance set to 3130 degrees Kelvin and shot in RAW