If there is one filter that is a must have for all of my lenses it is a Circular Polarizer. The effects of most other filters can be quickly recreated in post. There is still a need for an neutral density filter and to some extent a graduated ND filter. The latter can be reproduced in post but requires the blending of exposures through HDR or more manual methods.
Many people use Polarizers to darken blue skies and often to the point of surrealism. If this is your thing then have at it but I tend to use the filter for another purpose. The removal of reflection.
When using the polarizer for this purpose you simply rotate the end until you get the desired result. In the images seen here the one on the left was the non polarized version and you can see a lot of shine on the rock and wet vegetation. On the right you see the image with the polarizer turned to remove the reflection. The difference is dramatic. The reflection is gone and the colors are much more saturated.
The down side (sort of). When you attach a polarizer you will lose light and thus will have to make exposure adjustments. This isn't always a bad thing especially if you are taking photos of flowing water and the reduction of light is helpful for blurring the movement. How much light is lost? It depends on the polarizer. I am using the Nikon 77mm filter and lose about 2 stops when I attach the filter.
A quick word on filters in general. If you are going to put one on your lens for any reason be sure that it is a high quality filter, like those from Breakthrough Photography. Otherwise there could be image degradation and some can cause the camera to have difficulty focusing. After all why would you spend $2000.00 on a lens and slap a $15.00 piece of glass or plastic on the end. It defeats the purpose...