When working with meshes what you generally want is is quads, not triangles. The reason is that quads form nice closed loops.

Further more you would only allow quads to meet in 3, 4, or 5 edges per vertex. The 4 edges per vertex is the "normal" case that most of your mesh should have, it causes a regular grid of parabolic (euclidian) geometry with neutral curvature. Then patches of these meet in vertices with 3 edges to make it elliptic geometry with positive curvature or 5 edges to make it hyperbolic geometry with negative curvature.

You can ignore all of these and just randomly connect nearest neighbors to form triangles. But, then you still have only geometry, no useful topology, so not any better than a point cloud. A good topology is necessary for texturing, skinning, animation etc.