My question relates to the idea of satisfiability. If one can provide a model for which the following formula is true does it mean it is satisfiable?
$$\forall x\forall y\forall z\,((S(x,y) \land S(y,z))\implies S(x,z)). $$
Is the above formula satisfiable if I have the following model:
$A = \{a,b,c\}$
$S = \{(a,b),(b,c),(a,c)\}$?
If it is satisfiable, what would be required to make this formula valid in this model?
If it is not satisfiable what would be an appropriate model to make it satisfiable?
There are several terminological issues in your question, so (at the risk of appearing too pedantic) let me try to address those.
Given a sentence $\varphi$ and a structure $M$, we can talk about the sentence being true or false in $M$. To talk about true or false, we need both a sentence and a structure.
So the phrases "$\varphi$ is true" and "$\varphi$ is false" make no sense unless we implicitly have a structure $M$ around that we're interested in.
On the other hand, a sentence $\varphi$ is satisfiable if it is true in some structure. And $\varphi$ is valid if it is true in every structure. To talk about satisfiable or valid, we only need a sentence, we don't need a structure.
So the phrases "$\varphi$ is satisfiable in $M$" and "$\varphi$ is valid in $M$" make no sense. On its own, $\varphi$ is either satisfiable or not and either valid or not.
Now to answer your questions one by one, with the above in mind:
Yes, this is the definition of satisfiable. To prove that $\varphi$ is satisfiable, you can exhibit a structure $M$ in which $\varphi$ is true.
Yes, exhibiting the structure $M = (A,S)$ proves that $\varphi$ is satisfiable, since $\varphi$ is true in $M$.
This question is not well-formed ("valid in this model" makes no sense). In fact, the sentence is not valid, and you can prove this by exhibiting a structure in which the sentence is false. For example: \begin{align*} A &= \{a,b,c\}\\ S&= \{(a,b), (b,c)\}\end{align*}
A model doesn't make a sentence satisfiable. It's either satisfiable or it's not. You can prove that a sentence is satisfiable by exhibiting a model, like you did above.