I don't recommend using a UV filter on any lens, unless you are shooting from a boat, or maybe in rain where splashing water might get on the front lens element. Unfortunately camera stores love to sell people filters, because they can mark them up around %300, whereas they make extremely little on the lenses and bodies. Lenses, especially the L's are a lot tougher than you may think. It pays to be cautious, but remember that it is just a tool, even though it may not be inexpensive.
Some people recommend using a UV filter to protect against scratches, but truthfully, that is a bit silly to me. Lenses shouldn't scratch all that easily if handled properly, and if they do it would take a major scratch to affect your image quality. That is why it often pays to buy used lenses at reduced rates if they have a minor scratch or two on the front element.
It never made sense to me to buy an "L" lens, which is the highest optical quality that Canon offers, and to put another piece of glass in front of it in the form of a filter. When you add another piece of glass to your lens, you are actually degrading image quality and can run in to greater trouble with lens flare, which can give you that classic starburst effect, and, or, reduce the contrast of the overall image. In the days when we shot chrome film, we would often use a warming filter to boost the color temperature of the transparency a bit, especially in the bluish light of the early morning. But as Daniel mentioned earlier, we take care of color temperature adjustments in post processing now, or in camera while shooting.
A polarizing filter on the other hand is the photographers best friend. The classic books will tell you all kinds of rules for using a polarizing filter, but here is my advice. Instead of taking the time to mount the "PF" to your lens, try holding it up to the scene and turn it with your hands to see if it is delivering the desired effect, or doing anything at all. If you like what you see, then put it on the lens and compose your shot. This will save you a lot of time over mounting the filter then deciding that it doesn't do anything. Also it pays to remember that a polarizer helps to reduce glare on glass/windows, some metals, and some paints on metals. Your best bet is to always just look through the filter and see what it is doing for you. Be careful when choosing a PF, and check it on all of your lenses, sometimes PF's will vignette, due to their length (or thickness). Remember too that a PF on full polarization will reduce the light hitting your film, meter, or digital sensor by ~1 1/2 stops. This can be useful when you need to attain a slower shutter speed to get the classic silky flowing water affect.
Which "L" lens did you get and which body are you running it on?