All the shapefiles necessary can be found here. These are shapefiles of lakes, and rivers. Downloading one shapefile as an example should be enough to get me on the right track.
The nuts file can be downloaded here.
What I want is to calculate the distance from each possible point in one nuts3-polygon to the nearest river. If all the distances are calculated from each point, I want to determine per nuts3-polygon what the average, minimum and maximum distance is to the nearest river, together with the standard deviation (if possible).
My paint skills are probably not the best ever, but the idea is something like this picture below (for each of the dots). Then I would calculate the average of all the brown lines that originates from a point in the polygon.
I can work a bit with R and ArcGis. Online I do not find a solution that matches what I am looking for.