Q: What filters should I buy?
A: Circular Polarizer (CPL), Neutral Density/Graduated Neutral Density (ND/Grad ND). Maybe UV.
The best advice when buying a circular polarizer or neutral density filters is to buy it for the largest lens filter size. Most Nikon lenses have a maximum filter size of 77mm, so buy it for that size (there are exception, but the common size is 77mm). Then you can buy step-up ring which takes the 52mm, 62mm, 72mm thread up to 77mm allowing you to use the larger filter. Good filters can be expensive so rather than several mediocre filters to cover your various lens sizes, you can carry around one expensive filter with step up rings. The step up rings are incredibly cheap and are much easier to carry around then several filters.
This is one that you don't want to go cheap on. Stick with either a Hoya or a B+W. This filter makes the skies appear as a darker blue, makes whites pop out a little more and cuts down on reflections from the water or glass. Think of it like polarizing sunglasses, as they also darken the image by around a stop (depends on brand and type). They also eliminate sheen on plant life making the colours more vibrant. Everyone should have one of these.
This is like having sunglasses for your camera. It stops down the light allowing longer exposure times. This filter is very helpful when photographing waterfalls and landscapes during sunrises and sunsets. They have different levels of "intensity" such as 0.3, 0.6, 0.9, etc. etc. The best multi-purpose one would be a 0.6, which is also known as a "2-Stop" Filter. The 0.3's are really meant to be added to other filters and the darker you go, the longer exposure time will be needed. A tripod is required for the really darker ones, such as a ND 3.0!
ND filters may be either solid or graduated. Solid ND filters will reduce the light hitting the sensor overall. (For example, if you need to use a large aperture for shallow DOF on a bright day, or if you want to do a very long exposure to capture motion, and you can't stop down far enough.) Graduated ND filters are used for situations in which part of the scene is considerably brighter than other parts, a graduated neutral density filter can be used — for example, to prevent the sky from being overexposed in landscape photography. Graduated ND filters are usually square filters which are inserted into a filter holder, and may be either hard- or soft-edged, indicating either a sharp transition between the darkened area and the area, or a smooth one.
UV Filters and/or Protective filters
The original purpose of UV filters was to cut haze by filtering UV light. This is more important in film than it is in digital, since UV filtration is built into the camera in digital. Haze is created in the sky when light travels a long distance for those who typically stick to macro photography buying a UV filter is useless. If you want the protection buy a clear filter.
Warning with UV filters - If you notice a green colour spot/cast on your photos, it's caused by an outside light source diffracting against the filter element. Two ways to solve this issue: 1) remove the filter for the shot and 2) compose the photograph differently where the light source doesn't effect the image. Many consider this a deal breaker when working with UV as they could flare up at the worst moment.
Protective (or clear) filters do not have any special coding which can affect the image. These are used purely for protection of the front element.
Sky filters are another type of filter which enhance the blues in the sky. These filters make your photos slightly pinkish but are almost going extinct because of post-processing techniques.
For the question of whether UV filters or strictly clear filters for protective purposes are useful…see here
Warming filters add a warming red, orangy tint to your photos, often best for landscape. These filters are becoming rarer as more people are moving away from these types of filters but rather applying them in post-processing. Nearly all photo editing software have this feature.