The original definition of subset that you have is correct. A subset of R^3 is just any collection of points from R^3. For example, lines, planes, and arbitrary surfaces are subsets of R^3 because they are just collections of points.
I'm not sure what the definition of a neighborhood of a subset is, but a neighborhood of a point is just an open ball containing that point. Perhaps a neighborhood of a subset is a union of open balls over all the points in the subset.
The interior of a subset is the collection of points from the subset that have a neighborhood fully contained in the subset.