Correct me if I'm wrong, but is the difference between Euclidean geometry and non-Euclidean is that the latter includes the former but simply added the idea of doing measurements along the curved surfaces of multi-dimensional spheres and objects?

You can say that.

 

Non-Euclidean geometry deals with things that don't (globally) look like [math]\mathbb{R}^{n}[/math].

Yes.

 

Although I'd argue that it's the other way around, a geometry becomes Euclidean when you add the requirement that things are flat. So it also becomes Spherical when you add the requirement that things are sphere-ish or it becomes Hyperbolic when you add the requirement that things are sort of curved in that weird saddle shape thingymajig.

 

Point being, if you don't mention the parallel postulate that makes a geometry Euclidean, and you don't mention any of the alternative postulates that could make it Spherical or Hyperbolic or whatever, then you still have a geometry which isn't nearly as complete* as Euclidean or Non-Euclidean geometries but is still consistent and sort of useful because a theorem proven in that geometry will still be true in all the other geometries it spawns when you tack on that all important specification of curvature.

 

*does it make sense to talk about something being more complete or less complete?

You can't do much without dealing with parallels, though, directly or indirectly.

 

And per the OP, that might be more or less true, but I wouldn't describe it that way. A geometry is Euclidian when parallel lines neither converge nor diverge. Non-Euclidian is when they do. Talking about "curved" and "flat" space and whatnot is just shorthand for describing it in terms our Euclidian minds find easy to grasp, and models on the surface of spheres are just models, as a sphere (including its surface) is a Euclidian object.

The spherical and hyperbolic geometries are the two homogeneous non-Euclidean geometries. When we pass to more general Riemannian manifolds we have inhomogeneous non-Euclidean geometries, that is it may depend on exactly where you are on the manifold if the geometry looks Euclidean, spherical or hyperbolic.