Challenge 1: How to capture Moving Subjects in Low Light?

I am starting this new topic with the title CHALLENGE where I will be discussing the issues I faced multiple times while photographing and still looking for the answers... So it is your time to educate me if you have a solution... I am open for comments.

I always faced big challenges while capturing ongoing action in a stadium. Two big obstacles:
1. Objects are constantly moving... so the shutter speed should be fastest to freeze the action but it will not allow enough light to get in.
2. Light is anyway low... since they are far.. can't use the flash.                                                                      

About shots on the left: I took my new Nikon D5000 to the Circus... since it was my first time with the camera so I kept it on Auto and let camera decide the how to capture.

BUT RESULTS WERE NOT AS EXPECTED... RATHER BAD.

One of my friend said.. "DSLR hasn't done any better job than a Point-n-Shoot camera." Definitely I was not happy too with the blurry and grainy images because of high ISO values.

Here if the Exif for Circus shots:
Top left: 1/60s, f/7.1, ISO -
Top right: 1/3s, f/4.2, ISO 800
Bottom left: 1/13s, f/5.6, ISO 1000
Bottom right: 1/60s, f/4, ISO -

After some time, I went to Rodeo show in Fort Worth Texas and this time I was determined not to repeat the same mistakes again. So I kept the shutter speed as priority this time and made sure to keep ISO at lower numbers e.g. for most of them I kept 500.

BUT still got blurry images... I wanted to keep shutter speed faster say 1/500s but everything was coming dark though exposure bias was set to +2.7 step which is on the brighter side.

Here are the Exif for Rodeo Shots:
Top left: 1/8s, f/5.6, ISO 500
Top right: 1/20s, f/5.6, ISO 500
Bottom left: 1/20s, f/5.6, ISO 500
Bottom right: 1/60s, f/5.6, ISO 800

So my question here is: 

Do we have any option of correcting this issue other than buying an expensive DSLR with high ISO sensitivity or buying advanced lenses which can let more light in... even in low light conditions?

Also please let me know if I could have done better in the camera settings to get the appropriate results :)