I have purchased a Nikon D5000 camera with 18-55 from eBay.
The lens focuses automatically, but should the lens also zoom automatically or is this controlled manually by the zoom ring.
No, the lens do not zoom automatically.
Almost all SLR zoom lenses are zoomed using the zoom ring, very few have motorized zoom mechanism.
Every SLR and DLSR lens that I know of has manual zoom. This is because manual zoom gives you a higher degree of control than motorized, and also includes less complicated mechanisms in the lens and camera body, reducing the number of things which can break, as well as reducing the weight of the (often already very heavy) camera components.