To start with I am only considering two-dimensional space.
This is a list of assumptions and definitions from my simple perspective. Never quite sure what axioms are so I am not worrying about them.
A position in space, but does not occupy any space.
A flat surface that extends unlimited in depth and length, but has the same height. All points in a 2-D world lie on this space.
A collection of points all joined together. It indicates position but has no thickness.
A straight edge
A physical item that has one edge that when placed on another straight edge is in perfect contact with no gap between them wherever they are placed in contact.
Line that when placed against a straight edge is in perfect contact with no gap.