half a sphere
Technically, there's no shape with no surface and no vertices, but a sphere is close. It has no vertices but has just one surface.
A cone.
If you are a cube, one of the sides is called a face. Same with a cone, the flat part is a face.
A cone would fit the given description
An ovoid (egg-shape) is one possible answer. A smooth blob (to use a very technical term!) is another.
none!all shapes are flat
A shape with 0 vertices and 2 flat surfaces is a cylinder. The two flat surfaces are the circular bases at the top and bottom, while the curved surface connecting them does not count as a flat surface. Other than the cylinder, a shape that fits this description is a circular disk, which also has no vertices and one flat surface, but when considering the top and bottom of the cylinder, it effectively has two flat surfaces.
The shape you are describing is a cylinder. A cylinder has two flat circular faces (the top and bottom) and one curved surface that wraps around the sides. It has no edges or vertices where flat surfaces meet, as the curved surface is continuous.
Hemisphere
Technically, there's no shape with no surface and no vertices, but a sphere is close. It has no vertices but has just one surface.
cone
a cone has one flat surface one vertex and no edge.
A tetrahedron
It has a flat base, a curved surface area, a circular edge and one vertex.
a cone.
Circle
There can be no such shape. If there were a shape with n flat surfaces, then if you sliced off one vertex, you would have a shape with n+1 flat surfaces. So it is always possible to add one more flat surface.