Concrete examples of group actions.

You ask, in a comment, for a non-obvious action on $X=\{1,2,3,4\}$. Let me give you, instead, a non-trivial action of $S_5$ on $X=\{1,2,3,4,5,6\}$: it is given by an homomorphism $\phi:S_5\to S_6$ such that \begin{align} (1,2)&\longmapsto(1,2)(3,4)(5,6) \\ (1,2,3,4,5)&\longmapsto(1,2,3,4,5) \end{align} You should check that this homomorphism is injective. In fact, you should find all ways in which $S_5$ can act on this $X$.

If you change the numbers $5$ and $6$ and look for examples, you'll have lots of fun.

The dihedral group $D_8$, which is generated by elements $a$ and $b$ with the relations $a^2=1$, $b^2=1$, $(ab)^4=1$, acts on two coins resting side-by-side on a table, as follows: $a$ flips the coin on the left (so if it was heads, it becomes tails, and vice versa), and $b$ swaps the two coins (so the one that was on the left is now on the right, and vice versa).

Consider a cube in 3d space. There is a set of rotations of space that map the cube back onto itself - for example, rotation by 90 degree around an axis that cuts through two opposite faces in their middle. This set is in fact a group.

How does this group act on the set of 8 vertices? Name the vertices in some way and try to list the various permutations. Which group elements leave a vertex fixed? Which group elements leave two vertices fixed? Which vertices can those be? What if we consider the action on the edges of the cube instead of vertices? The faces? The main diagonals? What if we replace the cube by some other platonic solid such as an octahedron?

I hope this is less rote and abstract than the examples you saw so far.

It's hard to give examples without knowing a little about what you know and what you're comfortable with. If you know something about graphs, you can draw several small graphs and consider their automorphism groups - the different ways you can map their vertices to themselves while respecting the edge relationships. Or if you know some linear algebra and finite fields, consider a vector space of dimension 3 over the field $\mathbb{F}_2$ with two elements. How many subspaces of dimension 1 are there? What group naturally acts on those subspaces? How does the same group act on subspaces of dimension 2?

You also asked "How does a three cycle permute two elements?" Well, the 3-cycle $(1\, 2\, 3)$ maps 1 to 2, 2 to 3 and 3 to 1. Therefore, it maps the set $\{1,2\}$ to the set $\{2,3\}$, and the set $\{1,4\}$ to the set $\{2,4\}$. Can you take it from here?