Classification. You do not specify what classification (what is your equivalence relation?)
Topological classification is due to Kerekjarto. A reference is given in the answer of Richard Kent. Complete conformal classification is hopeless.
There are many different notions of hyperbolicity. They all coincide in the simply connected case.
a) Universal cover is the disc. (All Riemann surfaces, except tori, sphere, plane and cylinder
are hyperbolic in this sense).
b) There is no Green function.
c) There is no positive harmonic function
d) There is no bounded analytic function
e) There is no analytic function with finite Dirichlet integral
f) And so on.
In 1950-s there was large area of research called Classification of Riemann surfaces.
The main subject of this research was establishing the relations between b)-f) and
other similar properties, and finding criteria for concrete surfaces to satisfy b)-e).
You can find a lot of results of this sort in the books of Tsuji,
Potential theory in modern function theory, Nevanlinna, Uniformisation,
and Ahlfors and Sario book on Riemann
- Hurwitz automorphism theorem is not true. There are many open surfaces with a rich
group of automorphisms.
On your last remark. The subject is out of fasion, so modern books do not treat it.
There are just too many Riemann surfaces to have an interesting classification of
all of them. So the research is concentrated on various special classes that have
some applications. For example, on hyperelliptic surfaces (those obtained as a 2-sheeted
ramified covering of the plane); Such surfaces occur in mathematical physics.
Another interesting class which is studied is Abelian coverings of compact surfaces.