Focal length is focal length, regardless of sensor size or whether the lens is a zoom lens. If you have tried your kit lens at 35mm and 50mm, then the framing will pretty much be the same with prime lenses of those focal lengths. Prime lenses will offer a couple things your zoom lens does not, however.
For one, they should offer better quality, as prime lenses can be constructed to perfectly project the clearest image possible for the given focal length. There are degrees of optical quality within a given prime focal length, as a higher end lens will usually use better materials and lens elements. Generally speaking, though, primes offer better quality.
Second, prime lenses usually provide much wider maximum apertures. A 50mm prime can come in anywhere from f/1.8 through f/1.2, and older manual focus lenses may even be found with maximum apertures as wide as f/0.95. Wider apertures can be more difficult to use at times (due to extremely thin DOF), but they can provide some truly fantastic bokeh.
Regarding focal lengths on a cropped sensor, there is a fairly nice correlation between the 35mm and 50mm lenses and ideal portrait focal lengths on full-frame sensors. Nikon cameras have a 1.5x sensor crop factor. That means that a 35mm lens on an APS-C body "behaves as" a 52mm lens would on a full-frame body, due to the difference in the field of view captured by the APS-C sensor. The 50mm lens on APS-C behaves as a 75mm lens would on full-frame. As an additional lens, 85mm lenses behave like a 134mm lens on APS-C. These focal lengths fit pretty well with the ideal portrait focal lengths on full frame cameras, which include 50mm, 85mm, and 135mm.
It should be noted that 50mm lenses on 35mm film/full frame sensor produces a field of view that is very similar to the field of view of the human eye. The actual focal range for that falls between 45mm and 55mm.
So, given all of that...you can make the proper decision based on what you really want to capture. If you want to capture shots that have a relatively "normal" perspective similar to how the human eye sees, you might want to grab the 35mm lens. On a cropped sensor, it would behave like a 52mm lens. If you want a narrower field of view with smoother background blur, a 50mm or 85mm lens would give you that deeper DOF and narrower field, similar to 85mm and 135mm lenses on full frame, respectively. Finally, if you want to capture a wider field of view than the human eye, or want to get really close and capture a lot of perspective, you could get a 24mm lens, which, intriguingly, behaves like a 36mm lens on full frame.
The beauty of primes is their field of view, or effective focal lengths, are easy to translate between APS-C and Full Frame. If you take the common prime focal lengths of 14mm, 24mm, 35mm, 50mm, 85mm, and 135mm as used on full frame, you can pretty much simply "shift" them up by one place to arrive at the effective focal length ranges on APS-C. The exact focal lengths on APS-C, for reference, are: 21mm, 36mm, 52mm, 75mm, 134mm, 202mm. So long as you understand that about 50mm is the same field of view as "normal" perspective, it is fairly easy to determine which focal length to use to get the field of view/perspective you want in your photographs.