Visualizing Orbits and Transitivity I'm having trouble "visualizing" orbits and transitivity.  
So starting with an example, if I have the group $G=S_4$ acting on a non empty set $A=\{1,2,3,4\}$ I know that for example, the stabilizer of the element $2$, represented $G_2$ is isomorphic to $S_3$, because intuitively, if $2$ 'stays put', the other three element $1,3$ and $4$ can be permuted and thus there are 6 permutations of $\{1,3,4\}$.  Thus I know 
$$G_2=\{1_{S_4},(1 3),(14),(34),(134),(143)\}$$
Now I'm trying to understand orbits.  Orbits are equivalence classes.  Basically, I think that I'm looking at all 24 permutations acting on an element in $A$.  So, if $\sigma\cdot i=\sigma(i)$, I'm looking for all permutations that send any element to a particular place in the permutation.  So, the orbit of $S_4$ containing 2 would then be 
$$\{(24),(234),(1342),(1423),(13)(24),(142)\}$$
I'm pretty sure this is wrong.  These are elements of $S_4$ while my action is suppose to be into the set $A$
How do I visualize the appropriate orbits?  How can I view the action that is occurring in my mind so that these misunderstandings don't occur?  The question regarding transitivity also holds, since I need to understand orbits before understanding transitivity.
 A: 
Now I'm trying to understand orbits.

Great! Let's find an example that will illustrate what orbits are. Note that the usual action of the permutation group $S_4$ on the set $A = \{1,2,3,4\}$ is, orbit-wise, rather boring. This is because, for example, the orbit of $2$ under $S_4$ is
$$
\text{orb}(2) = \{\sigma (2) \,|\, \sigma \in S_4\} = \{1,2,3,4\}.
$$
In other words, there is only one orbit in $A$ under the action of $S_4$.
Now for an interesting example. Let $G$ be the group of rotation acting on the plane $\mathbb{R}^2$. The orbit of the point $0 = (0,0) \in \mathbb{R}^2$ under $G$ is simply $\{0\}$. However, the orbit of a non-zero point $x \in \mathbb{R}^2$ is the circle in $\mathbb{R}^2$ centered at the origin and containing $x$. (Can you see why?)
In this case, the set of all orbits is precisely the set of all circles centered at the origin, together with $\{0\}$. These circle are uniquely determined by a non-negative radius, and therefore the set of all orbits can be identified with $\mathbb{R}_{\geq 0}$.

How do I visualize the appropriate orbits?

I believe you ask how to visualize them in your mind and get an intuitive feeling for it. You can think of a group $G$ acting on $A$ as a set of invertible transformations on $A$. For instance, $A$ could be bread dough, with $G$ acting on $A$ by folding (or careful unfolding). If you place a seed in the dough, all the places the seed can go by folding and unfolding is precisely the orbit of the seed under $G$. 
If you always fold the dough in the same upward direction, then the seed won't be able to move left and right and will be contained in a plane. 
If you add more transformations to $G$ and mix the dough like crazy, then the seed will be able to go everywhere and you'll only have one orbit.
In short: The orbit of the seed $s$ under the action of $G$ is the set of all the places where $s$ can end up after being pushed around by the transformations in $G$.
