You are probably looking for something light and accessible. I would stick with spherical geometry as it is easier for students to conceptualise. Discrete spherical geometry makes it even easier and more hands-on.
Introduce basic graph theory and use Euler's formula to calculate the possible regular polyhedra (Platonic and Archimedean solids) which are topologically equivalent to a sphere. Spherical geometry can be studied from these polyhedra. You will also encounter 2 infinite series.
Draw graphs of some of these polyhedra and create nets to construct physical models. Some graphs are quite difficult, but can be made easier if you start at a vertex for the tetrahedron and a side for the most of the others. You want to finish drawing at a side, not a vertex. Practice drawing them yourself first. Grow the graph symmetrically around the starting figure. Look for forced lines (one line left at a vertex) and keep track of filled vertices (maybe draw a circle around them when they are full). Don't worry about curved lines, especially around the outside of the graph, as curves don't matter topologically.
Notice that the graphs of spherical surfaces become highly distorted. Draw polyhedra on a spheres (oranges/mandarins). Discuss strategies that map makers have used to address the distortion of spheres projected onto flat surfaces. You could peel and flatten an orange skin as a demonstration. Students can do it outside during recess.
Calculate the angle defect of each vertex of a polyhedra. Use Descartes theorem to link to Euler's characteristic.
Look at possible symmetries of a sphere. Discrete symmetries (of Platonic solids) correspond to discrete groups, another interesting and accessible topic. Continuous symmetries correspond to only a few Lie groups. You can make a hierarchy of symmetries based on experimentation on just Platonic solids.
Find underlying similarities between dual pairs of Platonic solids based on possible symmetries. Or transform graphs of any solid to its dual.
You can do group arithmetic on solids. Some group arithmetic is commutative, some is not! The single rotations of a prism compose with nice modular arithmetic, but see what happens when you allow a flip. Your students might even handle some simple problems using the 3 rotations of a cube. Keep in mind the full state of a cube with no reflection is the front face and orientation of the face - up/down/left/right - leading to 24 possible states, so you will not want to create a full Caley table, just do some simple arithmetic. Be very specific on your rotation directions.
Decorate solids to demonstrate the discrete spherical symmetries.
Extra for experts: create graphs and maybe nets of (parts of) regular discrete hyperbolic surfaces (full hyperbolic surfaces are infinite). Graphs will collapse into something like a Poincare disc rather than exploding like a sphere. Will take some creativity to make nets that can be stuck together.
I've used Geogebra for creating nets as it is easy to draw joined polygons, but it is not necessary for simply polyhedra. Some regular polygons to trace around maybe? It was often useful to draw graphs of surfaces first before doing the net.
I hope a few of these ideas are at the right level of your students.
