This is most likely what the user Sven was going to mention. It doesn't fully answer your question, but it's too long for a comment and I thought it might be useful.
So, let's go on to your first question. It is often difficult to find all the representations of a given finite group (at least practically). Often times one is presented with a group for which there are naturally equipped representations. For example, $S_n$ has the canonical one-dimensional sign representation $\text{sgn}:S_n\to\mathbb{C}^\times$, $Q_8$ (the quaternion group) has the natural matrix representation, etc. A common technique then for creating new irreps out of these canonically equipped ones is by tensoring them with one-dimensional representations. For example, there is the natural representation of $S_4$ by permuting the symbols $\{x_1,x_2,x_3,x_4\}$ of the free vector space $V=\mathbb{C}[x_1,x_2,x_3,x_4]$ and then passing to a three dimensional representation by looking at the invariant subspace of all elements of $V$ whose coefficients sum to zero (i.e. this is the standard representation). This restriction gives one a three dimensional irrep $\rho$, one can then obtain a different three-dimensional irrep by considering the tensor product $\text{sgn}\otimes\rho$. We know this is going to be irreducible. How? Because $\mathbb{C}$-representations are nice, and it suffices to check that $\chi_{\text{sgn}\otimes\rho}$ (the character associated to $\text{sgn}\otimes\rho$) is a unit vector in $\mathbb{C}[S_4]$. But, this is easy to check since $\chi_{\text{sgn}\otimes\rho}=\chi_\text{sgn}\otimes\chi_\rho$ and $\chi_\text{sgn}$ is degree one.
So, all in all, there is no fixed way to create all the representations of a given group (this should be morally true since the representation theory of a group gives a lot of the group's structure away, and so if finding the representation theory of a group is simple, so should finding the structure of the group--and this is definitely not so).
Lastly, to address your other question. I presume what you mean is, if you suspect that two representations $\rho:G\to \text{GL}(V)$ and $\psi:G\to\text{GL}(W)$ are unitarily equivalent how does one go about finding the actual transformation $V\to W$? Once again, I don't know any hard-set rule for this (although, I am less confident that there is no 'algorithm' for this--perhaps another user has a method). That said, the beautiful thing about $\mathbb{C}$-representations is that you don't really care what the actual transformation $V\to W$ is. In fact, for all intents and purposes it is not the actual representations $\rho,\psi$ which are important, but their characters, and being equivalent unitarily (or not unitarily) implies they have the exact same characters.
I hope this helps.