Depending on what time of day, the only way to achieve the long shutter speeds is with a neutral density filter. I shoot with a 9 stop, and Tom K has been getting great results with his 10 stop filter. This is what you're talking about, right?
Roberto has the idea, but as you start as a 'newbie' it is a perfect opportunity to build knowledge from square one. Basically you set your exposure for the overall scene, then step your shutter down in speed and compensate with stepping aperture up to maintain exposure. With digital you can fire a shot for every shutter speed stop and see if the blurring is how you want it. If you run out of aperture steps before the shutter is slow enough, then you have to add the ND filters as Roberto mentioned, up to whatever is required. Judging the blurring depends on water flow speed, but if you measure off say 3 paces along the flow, drop a leaf in the water and count the time taken to travel the 3 paces, then take the test shots and note the blur effects and keep a record of some approximate shutter speeds for water speed, you can get an idea of the future settings required to the effect you need. The shot displayed by Roberto is actually an excellent example of good effect, not overdone.
I'll throw in my two cents. If you don't shoot in bright sunlight, you may not need a super-duper netural density filter. I usually shoot flowing water at dawn, dusk, and in overcast light with a polarizer and I don't need a neutral density filter.
I set my camera on Aperture Priority. I bump up the aperture value until I get a corresponding shutter speed long enough to blur the water (1/4 second to a few seconds usually works out fine). On an overcast day, when I shoot with a polarizer on (which works to cut the light just like a neutral density filter), an aperture value between f11 and f16 usually will do the trick. So, in sum, bump up your aperture value, do a half-press on the button and then look to see what shutter speed your camera is giving you. Adjust aperture value accordingly.
Also, make sure you ISO is set at its lowest default level (usually 100). I higher ISO value will increase the shutter speed.
This photo was at aperture f13, which created a shutter speed of 6/10 of a second.
Daniel brings up a good point. There's quite a range of shutter speeds that work flowing water. My shot was in the neighborhood of 40 seconds I think. On a shot like Daniels he would have lost all the detail in his waves, whereas the SS he choose was a perfect mix of flowing water and preservation of detail. The only downside of going to an aperture of f/16 or above is that diffraction can cause a loss of sharpness in objects that aren't moving.
