Hi,
I suggest you work with Aspect map (spatial analyst). For example let us say you are walking north, that is if you are on an average moving north compared to your starting point.
Step1. Develop an aspect map of your study area.
Step2. convert your walking path to a raster (path-raster), but of same size and extent as your study area. Such that cells beneath path are valued 1 and all others valued 0.
Step3. Multiply your path raster with your aspect raster and get the output.
Step4. In the output raster if the values are between 90 degree and 270 degree, then you are walking uphill. if the values are between 270 degree and 90 degree, then you are walking downhill. A negative value in your output suggests that you are walking on a flat surface.
This is because you are walking north and the degrees are measured from north.
If you are walking in any other direction, just play around with the degree ranges to get a solution. Let me know if you dont follow my english.
regards