I don't know what your kit lens is. Possible that it is the 18-55mm zoom? Makes a bit of difference on a proper recommendation.
If that is your kit lens, then the complement to it is commonly the 55-200mm DX zoom. It is affordable, has VR, and gets decent enough reviews.
My first DSLR was a D80 that had an 18-135mm kit lens. The lens had a good zoom range, but otherwise not so good. I bought a used 70-300mm lens for just what you want to use it for. Mine was the older pre-VR version. I find it is a bit poor with contrast and is nearly as bad with chroma aberrations as the 18-135mm lens was. I still use it, have taken some nice pictures of wildlife using it, but always requires some extra attention in with the photo-editor to clean up the image. The VR version can be bought for reasonable prices now, but it generally gets criticized for "soft" images at the longer zoom ranges.
My all-in-one zoom now is a Tamron 18-270mm f3.5-6.3 VC PZD. I have been happy with this lens as it overcomes all of Nikon 18-135mm lens shortcomings. It can be used for wildlife, but it does not give the same frame field of view as the Nikon 70-300mm lens does at equivalent zoom lengths. It looks less zoomed-in, and it has to do with how focal length is calculated with the different geometries of the lenses. But it is an option.
So going back to 2011, here is an image I took with the D80 and 70--300mm combo. I think this is a JPG straight from the camera.
This was a favorite wildlife photo of mine for a while, and I always tried to work with it to make it a bit better. One of several post-processed copies...