It's pretty hard to answer this question when 'familiar' isn't defined further.
Every finite group (and thus every permutation group) has a composition series, which is unique in the sense that the length and composition factors of any two composition series are the same up to permutation and isomorphism. If you define familiar groups to be simple groups, then these quotients are the familiar pieces you're looking for.
Defining familiar as simple is a stretch, though. You'd be hard pressed to find someone who found the Held group especially recognizable.
The other problem is that two nonisomorphic groups can have the same composition factors. The factors do help you break the group into smaller, more recongizable chunks, but they don't tell you how those chunks interact.
There are so many variations on combinations of smaller groups that it is difficult to imagine how we could recognize them all without group presentations. Another way of breaking down a group into recognizable groups is by looking at its Sylow subgroups and seeing how they interact (this is local group theory). But for this, we need to understand $p$-groups enough to call them recognizable. My answer to this question should give you an idea of the magnitude of what we're dealing with just in the world of $p$-groups.
Take for example $\text{SmallGroup}(16,3)$. There is no other name for that group, as far as I know. It is isomorphic to $(\mathbb{Z}_4\times\mathbb{Z}_2)\rtimes \mathbb{Z}_2$, which shows us it can be split into those recognizable pieces. This decomposition uses semidirect products, however, which is basically the same as the relations / presentation concept we are trying to avoid.
So in summary, there are many a lot of different ways to divide a group up into pieces that are as familiar as possible, to gain understanding about it and how it works. However, to achieve a full description of the group, most of the time we need to use specific relations in a group presentation to show the way those pieces fit together.
Yes. To any group $G$ (and choice of generating set $S$) you can associate its Cayley graph, which has a vertex for each group element $g$, and an edge between the vertices corresponding to $g$ and $gs$ for each $s$ in $S$. The left action of $G$ on itself corresponds to rigid motions of the graph. This graph is finite if and only if $G$ is a finite group.
If you know a little more topology, a corollary of Van Kampen's theorem is that every group $G$ is the fundamental group of a 2-dimensional CW complex $X$, so in particular the group $G$ acts by deck transformations on the universal cover $\tilde X$. It even turns out that every finitely presented group $G$ is the fundamental group of a 4-dimensional topological manifold. In the same vein, Eilenberg and Mac Lane gave a "functorial" construction of a (typically huge) geometric object $BG$, an example of what they term a $K(G,1)$—a space whose topology is in some sense completely determined by $G$, its fundamental group. This allows one to use methods from algebraic topology on even finite groups.
ETA: The representation of infinite, discrete groups as distance-preserving transformations of geometric objects is a central concern of Geometric Group Theory! Meier's Groups, Graphs and Trees or Clay and Margalit's Office Hours With a Geometric Group Theorist make excellent introductions to this field.
Best Answer
It's good that you keep symmetry groups and symmetric groups well separate, they each use symmetry in a slightly different sort of way!
Classically, things like parabolas were considered symmetric: They have an axis of symmetry and reflection across that axis leaves the "footprint" of the parabola unchanged (even though individual points may get shuffled around). Regular triangles, squares (regular polygons in general), circles, and tilings (like a floor tile, or brick patterns) are all two-dimensional objects that have a lot (OK, usually at least a "nontrivial amount") of symmetry: they "look the same" from several different viewpoints.
Eventually, mathematicians formalized a symmetry to mean an isometry (function that preserves the distance between any two points) of Euclidean space that again leaves the "footprint" of (set of all points comprising) the object unchanged, while potentially shuffling the individual pieces. This is what Bye_world is referencing in the comments, and is really a pretty radical way to think about things!
The full set of symmetries of an object do form a group: The composition of symmetries (which, remember, are really just special functions from $\Bbb R^n$ [or related spaces] to itself) is yet another symmetry. Function composition is associative. The identity map is the identity symmetry, and it doesn't do anything. Finally we can "undo" any symmetry; for example performing a counterclockwise rotation of $28^\circ$ to undo a clockwise rotation of $28^\circ$, or translating in the opposite direction to undo a translation, etc.
This is why semigroups aren't really a very good language to talk about symmetry.
We don't have an identity. We can't be guaranteed that we can just "do nothing" to the object in consideration. Not only is this not-so-good (TM) because doing nothing is nice, but also because
We aren't guaranteed inverses -- let alone the ability to define them, if we can't be sure we don't have an identity element! This really doesn't get along well with our picture of symmetries of an object: Two viewpoints of an object aren't really "the same" if we can't switch freely between them, but only from one to the other, and not back: We need inverses, and we need an identity to talk about inverses.
Sidenote: More generally, we have automorphisms of mathematical objects. These are maps from an object to itself that preserves some essential structure; a more abstract version of symmetry. You've probably encountered group automorphisms, or graph automorphisms. I personally tend to think in the language of "symmetries of a graph" rather than "automorphisms of a graph" unless I need to be fancy for some reason. This is distinct from a symmetry in the "isometry" sense, although the line gets a little blurry, namely because we like to draw ( = "realize") graphs in $\Bbb R^2$, and then the notions sometimes coincide!
Again, automorphisms come together to naturally form a group, and "automorphism" is a term that can be applied to a surprisingly large number of kinds of structures. This is the sense in which groups are the language of symmetry: If you have things that "act like" (or are) symmetries, then they form a group!
Polynomials are another place where "symmetric" is somewhat commonly used: There are things called symmetric polynomials (or more generally, symmetric functions), like $f(x, y) = x^2y + xy^2$ whose values don't change when we permute the variables; so $f(x,y) = f(y, x)$ above. I forget any surprising places these show up, but I know that Newton did some work with them, predating Galois and the formal definition of a group.
At any rate, I had suspected this is where the term "Symmetric Group" got its name (since the permutations of the variables do indeed form symmetric groups), and this answer confirms it, by quoting a MathOverflow post quoting the pioneer Burnside!