If you just want bokeh for bokeh's sake then you can achieve this with pretty much any lens and any type of camera, even a tiny sensor compact, by focusing extremely close. Depth of field diminishes very quickly with focus distance, so much so that it becomes a major problem with macro photography getting a non blurred background (or subject!)

However this approach leaves you only able to shoot very small subjects. A very common subject for blurred backgrounds is portraits, so this is the case I'm going to consider.

With a standard kit zoom your aperture is limited to f/3.5 at the wide end and f/5.6 at the long end. Conventional wisdom states a larger aperture will give you shallower depth of field and more blur. However longer focal lengths enlarge the background blur so which should you use?

I will answer this with the help of a giraffe named Sophia who stands about 4 feet tall and has a head roughly the right size for a realistic focus distance. It can be difficult comparing blur at different focal lengths so I have chosen a Christmas scene so the points of light show the blur radius clearly.

Here is what our scene would look like with an 18mm lens at f/3.5 on an APS-C sensor:

And now here is how the same scene would look with at 55mm f/5.6 when the camera subject distance is changed to maintain subject size. It could be argued that the camera should have stayed put, but in either case the same number of variables change, this way better reflects actual usage.

Not only is the background less cluttered due to the narrower field of view, but the blur is greater if you look at the Christmas tree lights.

So you can get a degree of blur from your kit lens with a real subject, when used relatively close. The background is still recognisable which can be a problem with less attractive settings.

The best thing you can do for Bokeh is to invest in your system's 50 f/1.8 lens. This combines the longer focal length effect seen above with a significantly wider aperture. All the major manufacturer's offer such a lens and due to it's symmetrical construction they can usually be snapped up for only $100-150. The same scene with a 50 f/1.8 would look something like this:

Now comparing this to the previous shot, the bars of the chair in the bottom left corner are gone, as are any details in the tree's leaves. You can be assured that any distracting details in the background will be gone when using this lens wide open.

If you're really into blurred backgrounds and want to push it as far as you can upgrading to a full frame camera is an ever more tempting option as prices come down. Just for fun here's what you can expect from a full frame camera and 85mm f/1.2 lens:

The background is now so blurred that the mirror box is acting as a second aperture and producing the strange cropped shaped highlights.