I'm confused about camera lenses?

I got a Nikon D3000 last year, and have been using the lens it came with (18-55mm). But now I'm getting annoyed by the lack of zoom, I have to crop the image after to get the zoom I'm looking for. Having looked on lots of buyer guides I think I want a telephoto zoom lens - maybe up to 300mm? I'm not really sure what to be looking for, or what's compatible with my camera. What's a good, yet cheap telephoto lens I could get?

You are looking for the new Nikkor AF-S 55~300MM DX or the older 55~200MM DX. They are affordable.

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Or may be you are looking at a close up or macro lens then get the AF-S 85MM DX Micro. Affordable too.

If you concerned on cheap telephoto lens, you may consider Sigma 70-300 or Tamron 18-270mm.
Of course you have to sacrifice some picture quality compared to Nikkor lenses but it's worth the price tag.

The 55-200mm VR Dx lens might be a good choice for you. My son has a D3000 with this lens (and the 18-55mm).

You might wonder why go with the 200mm vs the 300mm. It is all just preference. There really is not a huge difference magnification-wise between 200mm and 300mm, and some prefer the 200mm as it is lighter, and to some extent, lower power zooms generally offer better optics (which is not always the case).

And the 55-200mm VR is less expensive.