What 3d shape has more than one vertex?

Possibly a cone?

No, A cone is one of the few solid objects with one vertex.

A pyramid (which includes a tetrahedron) has 4 or more.

A prism (which includes a cube) has 6 or more (but even numbers only).

Other regular solids are octahedron (6 vertices), dodecahedron (20), icosahedron (12).