Will The Nikkor 55-300 High Power Zoom lense work with the Nikon D3000?

Will this Lense

http://www.nikonusa.com/...ED-VR.html

Work with my new Nikon D3000

Yes, all lenses made for Nikon cameras with an internal autofocus motor will work perfectly with your camera.
Nikon calls it "AF-S", be aware that that's not the same as "AF". Other brands use different names, such as HSM in case of Sigma.

Of course it will