^Polarized is different. A UV filter just takes out the UV range. A polarizer actually spins and you rotate it to get the right angle to take out the range of light.
If you shoot too slow without a tripod you will get a blurry picture (everything.) I'm not sure what happened with your photo (can you post it here for us to look at (EXIF information as well?) I'm guessing you didn't hold steady with the slow shutter speed and moved a bit with the shower motion and "froze it" and blurred the rest.
Skies. What camera are you using? What you are looking for is the exposure compensation/bracketing feature on your camera. Most DSLRs do this by pushing a couple of buttons to move the exposure either up (overexposed) or down (underexposed). You'll want to underexpose a bit to avoid washed out skies. Depending on what your camera is doing for determining exposure (middle, spot, overall average) you will need to either spot on the object you want properly exposed and push the shutter button halfway down then move back to where you want it in the frame and push the rest of the way. This would be so much easier in person.
I have no examples of doing the water thing. This is the only example I could find right now of using motion in a photo. Keeping the shutter reasonably slow (1/50) to keep the grass blowing in the wind.)
Exposure: 0.02 sec (1/50)
Aperture: f/11
Focal Length: 17 mm
Exposure Bias: 0/6 EV