One way to visualize higher dimensional objects is to draw a picture
of their intersection with 3 dimensional hyperplanes. For example
if we intersect a hypersurface X in projective space with a linear
space which does contain it we will obtain a hypersurface in the
linear space. After a projective linear transformation we may take
the linear space to be x_k=x_[k+1}= ... = x_n=0. The intersection
of this with the hypersurface F(x_0...x_n)=0 is clearly the hypersurface
F(x_0,...,x_{k-1},0,0,...0)=0.
The Veronese and Segre varieties are rarely hypersurfaces. The
only Veronese variety which is a hypersurface is the
rational normal curve in the projective plane (generally a Veronese
variety is an embedding of projective n-space in projective N space
where N exceeds n+1 except for the case of the conic in P^2). The only
Segre variety which is a hypersurface is the case of the product of
the projective line with itself embedded as a degree 2 hypersurface
in three space.
The intersection of a hyperplane with the image of a Veronese map
P^n->P^N is the image of the hypersurface in P^n obtained by composing
the equation of the hyperplane with the Veronese map. Thus a 3 dimensional
slice of a Veronese variety will not be a surface in 3-space since it is
the image of the intersection of at least n hypersurfaces in P^n which will
generally not be a surface. For example, the Veronese surface in P^5 will
intersect each dimension 3 linear space in a finite set of points.
Similar dimension counting shows that any Segre variety of dimension at least
3 is not a hypersurface, and a three dimensional section of it is a curve
when the variety is the Segre threefold P^2 X P^1 and a finite set otherwise.