  Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian
systems with $2n-1$ independent constants of the motion, globally defined, the
maximum number possible. They are very special because they can be solved
algebraically. In this paper we show explicitly, mostly through examples of 2nd
order superintegrable systems in 2 dimensions, how the trajectories can be
determined in detail using rather elementary algebraic, geometric and analytic
methods applied to the closed quadratic algebra of symmetries of the system. We
treat a family of 2nd order degenerate systems: oscillator analogies on
Darboux, nonzero constant curvature, and flat spaces, related to one another
via contractions, and obeying Kepler's laws. Then we treat two 2nd order
nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere
and its contraction to a Euclidean space caged Coulomb problem. In all cases
the symmetry algebra structure provides detailed information about the
trajectories. An interesting example is the occurrence of ''metronome orbits'',
trajectories confined to an arc rather than a loop, which are indicated clearly
from the structure equations but might be overlooked using more traditional
methods. We also treat the Post-Winternitz system, an example of a classical
4th order superintegrable system that cannot be solved using separation of
variables. Finally we treat a superintegrable system, related to the addition
theorem for elliptic functions, whose constants of the motion are only rational
in the momenta, a system of special interest because its constants of the
motion generate a closed polynomial algebra. This paper contains many new
results but we have tried to present most of the materials in a fashion that is
easily accessible to nonexperts, in order to provide entr\'ee to
superintegrablity theory.
