Which lens would be compatible with my nikon d3000 Digital SLR?

I'm looking at the Nikon AF-S VR Zoom-Nikkor 70-300mm Zoom Lens, because i need a lens which can zoom in far, i have seen by reviews that it is a great lens. The only thing i want to know, is if would work with my DSLR.
and if this lens doesn't work, which would with my camera that has a really good zoom?

Of course it will work. The whole point of buying a DSLR is that you can use other lenses. Your camera can use all current Nikon lenses. It will only auto focus with the AFs and AFi lenses but you can use the older lenses with manual focus.

It will work and I can tell you from long personal experience that it is a fine lens. For future reference, anything nikon af-s, sigma HSM or tamron built-in-motor will work.