Consider that a cone is what you get if you take a pyramid with a base formed by a polygon, and increase the number of polygon sides to a very large number. A little more formally, if we represent the number of sides of the base polygon with n (we'll call the polygon an n-gon, following the form of a pentagon, a hexagon, etc), then we say that a cone is the limit of our n-gon pyramid as n goes to infinity.
The number of sides of an n-gon is n, by definition, and the number of vertices is also n. As the base of the pyramid, the n-gon is one face.
We add one more vertex at the top of the pyramid, so we have V = n + 1.
We draw n more edges, from the new vertex to each of the n vertices on the n-gon. So, we have E = n + n.
Those new edges define n new faces, one between the vertex and each of the n sides of the cone, so we have F = 1 + n.
Then, we want the limit as n goes to infinity of V - E + F.
lim (V - E + F)
lim [(n+1) - (n+n) + (1+n)]
The n's cancel out, and we're left with
which is simply 2.
You can do a similar thing with a cylinder, considering it to be the limit of a prism with an n-gon base as n goes to infinity.
A sphere is tougher to visualize, but you can consider it the limit of a regular n-hedron as n goes to infinity, and it will still satisfy V - E + F = 2.