I get asked this question quite often by people in my workshops and by people that meet me while I am out shooting. It used to be that you needed to keep an assortment of filters at the ready to better realize your artistic vision when shooting with film. In my opinion today you really only need 2 different filters.
I highly recommend a skylight or a UV filter for each of your lenses. This is a bit of a philosophical debate amongst most photographers. Some will tell you that nothing should be placed in front of the lens, period! I do not subscribe to this philosophy and I advocate either a skylight or UV filter for every lens. In my opinion I feel they offer a bit of protection from finger prints, water droplets, accidental contact with a nearby tree or plant, and curious fingers. I also like the fact that if I do get a bit of something on my lens while shooting I don’t feel so bashful about just wiping it off. Without the filter I feel like I need to do minor surgery before I touch the front element of my expensive lens.
Next up is a Circular Polarizer (CP) filter. This filter is great for anyone who shoot outdoors and wants to darken up the sky a bit, remove some unwanted reflection from a window or water, or 1 stop Neutral Density (ND) filter. The CP filter allows you to control the direction of the light entering the lens by rotating a front element until the desired effect is seen through the viewfinder.
All filters come in different sizes to match the tread size on the front of your lens with 58mm and 77mm being quite common. As with lenses there are all different price points for even these most common types. I generally recommend getting a filter that matches your investment in your lens. It is silly to spend $1,000 or more on a lens and then put a $7 filter on it. For me I prefer the multi-coated filters and for my wide angle lenses I also get the “thin” filters as they reduce the chances of vignetting or darkening in the corners when shooting at the widest setting of the lens.