Some of the below repeats parts of other answers. Apologies for that...
The higher the ISO, and the longer the shutterspeed, the more noise you're going to get.
Some cameras are better than others. Cameras with larger sensors tend to have lower noise and better high ISO performance. A high end full frame 35 mm DSLR with good high iso performance would make a big difference.
Given your current setup, shoot with the fastest lens you can get, with the lowest ISO you can. You might try using a larger aperture.
The good news is that the noisy parts of your image are out-of-focus.
If it were my shot I would open the raw image in Photoshop, select the bird and the tree, and then invert the selection to select the sky and apply moderate gaussian blur to the sky. That would hide the noise quite effectively. You could also use the same technique to select the sky and then apply a variety of other noise reduction techniques, but for an out-of-focus sky, blurring would be easy, effective, and give very natural-looking results.
I downloaded your image and took the liberty of doing what I describe. Here is a 100% crop of just the eagle so you can see the effect:
I would NOT use global noise-reduction software, as that tends to destroy detail in the parts of the image you want (the bird and the trees.)