What zoom lens would be compatible with my Nikon D3000?

My husband recently got me a Nikon D3000 to ease me into DSLRs. I would like topurchase a zoom lens for it, but would like to know the best and most compatible lens for this particular camera. I would like as much zoom as possible without breaking the bank.

Ah, the beginners's reflex… Zoom, I need zoom! Tamron maken an 18-270 that is cheap.

Well, do you want quality or quantity? And you really have to tell me what breaking the bank is for you.
I'm guessing the bank isn't going to allow the 70-200 f/2.8 at $2500 or there about?
How about the Sigma at $1350 or so?
Or the cheap sigma at $1000?
Or the tamron 70-200 at $700 or so?
Still too high? We're down to quantity now, not quality. The tamron mentioned before is a decent walk around lens for someone who isn't into extreme tack sharpness and perfection.

The Tamron 18-270mm offers lovely versatility and convenience at the expense of image quality. The Nikon 18-55mm AF-S VR and the Nikon 55-200mm AF-S VR will offer the focal lengths that are likely to be most useful to you with better quality and minimum cost.
With the money you save, get a good Manfrotto or Cullmann tripod.

First of all: since the D3000 lacks an internal focusing drive found in older or more expensive Nikon models, you'll need a so-called G-series Nikon-lens (or G-series compatible if you buy a non-Nikon brand). G-series lenses have their own internal, auto-focusing drive.

That said, why not go for the excellent 55-200 VR tele zoom and possibly the 18-55 VR wide-angle to moderate lelezoom. They're cheap, but their optical quality is great and they have image stabilisation (VR = Vibration Reduction) which effectively reduces blurring caused by motion. They're not built to the exacting standards of lenses for professional use costing at least five to six times as much, but as long as you don't let them drop you'll be fine. I've been using these lenses for years without any problem, they're light and small, and produce great results. The 55-300 suggested above has a bit more reach, and while excellent too, is bulkier and considerably more expensive and because of its longer focal length, is more susceptible to motion blur when used at its longest tele setting under poor lighting conditions.