But the geometry in algebraic geometry is dictated entirely by the algebra. So how can you use geometry to help you with the algebra, when you have to do the algebra first to figure out what the geometry looks like? I don't get it.
Roughly speaking, geometric intuition suggests to you what should be true, and suggests a strategy for proving it; and algebra is how you carry out the proofs. For example, it's geometry that should inform your intuitions about things like intersections of varieties, but algebra that you use to actually rigorously define intersections and prove things about them.
Consider the "principle of continuity." If you intersect a circle and an ellipse, you'll generically get four points, although sometimes you might get two or one. If you intersect a circle and a parabola the same thing happens. In a perfect world, you might suspect that the intersection of two conic sections is always "four points," but this clearly isn't true for the usual definition of "four points." But if you broaden your definitions (count the points with multiplicity; count the complex points; count the points at infinity), you will eventually be led to complex projective varieties, where something like the "principle of continuity" holds: if you have two varieties that intersect in $m$ points and you nudge one of them or the other continuously, they will still intersect in $m$ points, if you count the points properly.
It's important to keep in mind that historically a major incentive for developing commutative algebra was to rigorize parts of algebraic geometry; some algebraic geometers (the Italian school) had begun relying too heavily on geometric intuition and had been ignoring special cases, etc. and commutative algebra was one way to fix their proofs. But the point is that they were doing algebraic geometry first! There is a long and interesting history here which I think it is very instructive to learn; you should try to find Dieudonne's History of Algebraic Geometry, as well as (for a personal perspective) Parikh's The Unreal Life of Oscar Zariski.
When I first saw the axioms for a group, I spent the next year trying to figure out why the heck anybody cared about groups (and frankly still only know this in a detached and academic way).
The group axioms are an abstraction of the notion of symmetry, and symmetry is a natural and beautiful idea, and symmetries are everywhere in mathematics. Perhaps you aren't acquainted with enough examples; it's hard for me to give more specific advice here without knowing what you find unsatisfying about groups. But you might be interested in Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things, as well as in Mumford, Series, and Wright's Indra's Pearls: the Vision of Felix Klein.
Best Answer
In the linear algebra of Euclidean space (i.e. $\mathbb R^n$), the consideration of subspaces and their orthogonal complements are fundamental: if $V$ is a subspace of $\mathbb R^n$ then we think of it as filling out "some of" the dimensions in $\mathbb R^n$, and then its orthogonal complement $V^{\perp}$ fills out the other directions. Together they span $\mathbb R^n$ in a minimial way (i.e. with no redundancies, i.e. $\mathbb R^n$ is the direct sum of $V$ and $V^{\perp}$).
Now in more general settings (say modules over a ring) we don't have an inner product and so we can't form orthogonal complements, but we can still talk about submodules and quotients.
So if $A$ is a submodule of $B$, then $A$ fills up "some of the directions" in $B$, and the remaining directions are encoded in $B/A$.
Now by itself this doesn't seem like anything new, or worth memorializing with new terminology, but often what happens is that one has a submodule $A \subset B$, and then a surjection $B \to C$, given without any a priori relation to each other.
However, if $A$ is precisely the kernel of the map $B \to C$, then we are (somewhat secretly) in the previous situation: $A$ fills out some of the directions in $B$, and all the complementary directions are encoded in $C$.
So we introduce the terminology "$\, \, 0 \to A \to B \to C \to 0$ is a short exact sequence" to capture this situation.
Since long (i.e. not necessarily short) exact sequences can always be broken up into a bunch of short exact sequences that are glued together, getting a feeling for short exact sequences is a good first step.
Of course, you should be coupling your study of these homological concepts with examples, e.g. short exact sequences arising from tangent and normal bundles to submanifolds of manifolds, all the important long exact sequences in homology theory (from algebraic topology), and so on; without these examples of naturally occuring set-ups of the "$A, B, C$" form described above, it won't be so easy to get a feel for why this concept was isolated as being a fundamental one.