Some filters you can live without, others you can't. It's as simple as that and anyone who thinks Photoshop and Lightroom can save them money and do the trick is cheating themselves.
Using the right filter at the right time will save you a lot of work in post processing, but it will also mean the difference between a good photo and a bad one. And one thing you should know by now is that no amount of post processing will help a bad photo.
With that said, here are the three filters we believe should be in all the camera bags in the world.
You just can't go shooting outdoors without one of these. Not literally of course, but you shouldn't. It is the one filter the effect of which cannot be replicated in post processing. Besides adding more contrast and making summer skies look a lot better, it will help your camera deal with something it can't on its own: reflections. The polarizer will help you shoot through windshields and water and it will reduce other reflections to a minimum. If you don't own any filters yet, start with this one.
Straight ND filter
It's essentially a piece of dark glass that looks a lot like welding glass. In fact, some photographers try using welding glass instead because it essentially does the same thing, it blocks light. Using a dedicated filter however will give you the right contrast and colors, not to mention more control. A straight ND filter is very useful on bright sunny days when you want to shoot at wide open apertures. It's also an indispensible tool for anyone shooting video with their DSLR.
A graduated neutral density filter is a piece of glass or plastic that is mounted on the front of the lens. Half of it is clear, the other half gradually becomes darker, hence the name. It's widely used by landscape photographers for evening out the exposure between sky and land.
It also helps preserve details and highlights and the photos will look a whole lot better than you HDR attempts.