Nowadays, there are some tools avaiable for visualizing algebraic surfaces.
Here are some of the tools I know:
surfex (with S. Holzer,
a tool which is quite easy to use, with many features
(sevearl surfaces, curves on surfaces, ...), it uses
surf (see below)),
surfer (with H. Meyer, C. Stussak and others,
very easy to use, and easy to install under Windows/Linux),
but much fewer features than surfex, it also uses
surf (see below)).
Some other tools:
surf (by S.
Endraß and others, difficult to use; therefore, see the tools mentioned above),
PovRay (a raytracer which can do much more
than drawing algebraic surfaces),
SingSurf (by R. Morris, usable directly in the internet,
the quality is not very high, though),
axel (by J. Wintz,
B. Mourrain, and others, currently only available for Linux, Mac).