It seems to me that your goal should be to give them some elementary results with proofs (so they get used to that type of thinking) but also some high-level, cool, and surprising results to whet their appetite for more. In studying planarity you can find many cool and surprising results. I took a course from Laszlo Lovasz on such topics a few years ago, and he put his lecture notes online. You can link to them from this site: http://www.cs.elte.hu/~lovasz/geomgraph.html

Highlights I think high school students could understand and enjoy (perhaps skipping some of the proofs):

• Section 1.1. Kuratowski's Theorem: A graph $G$ is embeddable in the plane if and only if it does not contain a subgraph homeomorphic to the complete graph $K_5$ or the complete bipartite graph $K_{3,3}$.
• Section 1.3. Steinitz's Theorem: A simple graph is isomorphic to the skeleton of a 3-polytope if and only if it is 3-connected and planar.
• Section 2.1. Unit Distance Graphs
• Chapter 4. Turning the edges of $G$ into rubber-bands and talking about when they find equilibrium in the form of a planar representation of $G$

Finally, Chapter 6 has some really amazing results:

• The Cage Theorem: Every 3-connected planar graph is isomorphic to the 1-skeleton of a convex 3-polytope such that every edge of the polytope touches a given sphere.
• Koebe's Theorem: Let $G$ be a 3-connected planar graph. Then one can assign to each node $i$ a circle $C_i$ in the plane so that their interiors are disjoint, and two nodes are adjacent if and only if the corresponding circles are tangent.