Is it safe to use Sigma lenses with my Nikon D3000?

I recently purchased the Nikon D3000, and I want a macro lens. It looks like the cheapest one is a Sigma lens, which takes good images. Would using a sigma lens on a Nikon camera damage the actual camera? If so, what cheap Nikon lens should I get for macro?

Yes, perfectly safe (provided it is Nikon fit, of course). Sigma are very highly rated manufacturers of lens for many marques of camera.

Since you have a motor-less D3000 you need to get a Sigma lens that has an auto-focus motor or you will be stuck in manual focus.