Points, lines, and planes are the three undefined terms of Euclidean geometry. A point has no dimension (just location), a line extends infinitely in one dimension, and a plane extends infinitely in two dimensions. All other geometric objects are defined in terms of these three building blocks, making them the axiomatic foundation of the entire subject.
Start with physical intuition: a point is a dot, a line is a taut string extended forever, a plane is an infinite flat surface. Then introduce the key postulates: two points determine a unique line, three noncollinear points determine a unique plane. Use diagrams heavily, emphasizing that drawings are imperfect representations of ideal objects. Practice identifying collinear and coplanar points.
Every subject needs a starting point — a set of primitive ideas that are taken as given and used to define everything else. In Euclidean geometry, those starting points are point, line, and plane. They are called undefined terms not because they are mysterious, but because defining them in terms of simpler objects would require even simpler objects, leading to an infinite regress. Instead, we accept their intuitive meaning and then build everything rigorously from them.
A point has only location — no size, no width, no area. In your earlier work with the coordinate plane, you graphed points as (x, y) pairs, and that intuition carries over: a point is a precise position in space, nothing more. A line is an infinite, perfectly straight, one-dimensional path. It has length but no width, and it extends without end in both directions. The dot you draw on paper and the segment you draw for a "line" are imperfect physical representations of ideal mathematical objects — the map is not the territory. A plane is a flat, two-dimensional surface with infinite extent in all directions and zero thickness.
What makes these useful is how they interact, captured in postulates (accepted rules without proof). Two distinct points determine exactly one line — there's only one straight path connecting them. Three noncollinear points (not all on the same line) determine exactly one plane — you need a third point off the line to nail down the plane's orientation. These postulates explain why a tripod (three legs, three contact points) is inherently stable while a two-legged stool is not: three noncollinear points fix a plane; two points only fix a line, leaving rotation free.
Intersections follow from dimension. Two distinct lines in a plane are either parallel (never meet) or intersect at exactly one point. Two distinct planes are either parallel or intersect along an entire line — never at just a single point, because the intersection of two planes is itself a flat object, and a flat one-dimensional slice of two planes is a line. A line and a plane either are parallel, intersect at one point, or the line lies entirely within the plane. These rules govern all of Euclidean geometry and everything built on top of it — segments, angles, triangles, and beyond all presuppose this foundational vocabulary.