There is a vast literature on the classification of finite linear groups over various fields. Over the complex or real fields, all finite linear groups are conjugate to subgroups of the respective unitary or orthogonal group, so as remarked in one of the comments above, studying finite groups of isometries in this context is the same as studying all the finite subgroups of ${\rm GL}(n,\mathbb{C})$ or ${\rm GL}(n,\mathbb{R}).$ As Richard Borcherds remarked, this soon becomes a complicated
problem. But strategies have evolved since the birth of representation theory to tackle the problem (for general fields) difficult as it is, in a systematic way. I'll discuss the real and complex cases. Generally speaking, we want to concentrate attention on linear groups which can't be described
in some "obvious" way in terms of linear groups in smaller dimensions. The first reduction, then,
is to concentrate on irreducible groups, those which leave no proper non-zero subspace invariant.
Maschke's Theorem tells us that no information is lost in the reduction. Another question,
for real representations, is what changes if we extend scalars to the complex field, where life
is generally easier. An irreducible real linear group may become reducible when the scalars are
extended to the complex numbers (this only happens when its character has squared-norm $2$ or $4$).
In each case, the real finite linear group is isomorphic to a finite complex linear group
in half the original dimension. So now I only speak of finite complex linear groups.
As remarked in someone's earlier comment, the next natural reduction is to the case of primitive
linear groups, those which (up to equivalence) be induced from linear groups of smaller dimension.
There are strong restrictions on normal subgroups of finite primitive linear groups. In particular,
the structure of primitive solvable finite linear groups is very tight, and is well-understood.
Having reduced to the primitive case (back to the general finite group), the next question
is whether the underlying module is a tensor product of two non-trivial modules of smaller
dimension. At this point, it may be necessary to take (still finite) central extensions of the group you started with. If there is a non-trivial tensor factorization, then we are reduced to questions
in smaller dimension. If there is no such factorization (even allowing for central extensions), then the structure of the residual groups is very restricted indeed. The given representation may
be "tensor induced" from a representation (of smaller dimension) of a proper subgroup. Tensor induction was introduced by Serre. If it can't be tensor induced from a lower dimensional
representation (again, even allowing for central extensions), then the only possibility that
remains is subgroup of a central extension of the automorphism group of a finite simple group
(containing all inner automorphisms). Many mathematicians, for example, Guralnick, Tiep, Zalesski,
have calculated (relatively) low dimensional complex representations of (central extensions of)
finite simple groups in recent years. My answer is therefore: yes, it is a difficult question,
but one which can be addressed systematically in any given case, and for which much hard-won
theory is available in the mathematical literature. Addendum: Just as it becomes impractical to list
all groups of a given finite order relatively soon, and we have to content ourselves with understanding the "building blocks", that is, the finite simple groups, so it is with finite
linear groups. There are three types of building blocks for finite complex linear groups:
a) 1-dimensional cyclic linear groups.
b) Finite complex linear groups $G$ of dimension $p^{n}$, for some prime $p$ and integer $n > 0$,
which have an irreducible normal $p$-subgroup $E$ (extraspecial of order $p^{2n+1}$ and
exponent $p$ when $p$ is odd; either extraspecial or the central product of an extraspecial
group of order $p^{2n+1}$ with a cyclic group of order $4$ when $p = 2.$). In this case,
$G/EZ(G)$ is isomorphic to an irreducible subgroup of the finite symplectic group ${\rm Sp}(2n,p)$.
c) Finite complex linear groups $G$ of degree $m$ which have an irreducible quasisimple
subgroup $S$ ( this means that $S = S^{\prime}$ and $S/Z(S)$ is a non-Abelian simple group).
Then $G/SZ(G)$ is a subgroup of the outer automorphism group of $S/Z(S)$.
The third type of building block naturally does not occur for solvable linear groups.

In both cases b) and c), the respective subgroups $E$ and $S$ are minimal subject to being normal,
but not central.