The following system of PDEs
$$\frac{\partial V}{\partial t}+c\frac{\partial V}{\partial x}=\gamma(U-V)$$ $$\frac{\partial U}{\partial t}=\beta(V-U)$$ can be nondimensionalized by change of the independent variables to $\xi=\gamma x/c$ and $\tau=\beta(t-x/c)$, where $c$, $\gamma$ and $\beta$ are constants. This will give
$$\frac{\partial V}{\partial \xi}=U-V$$ $$\frac{\partial U}{\partial \tau}=V-U.$$
I am curious as to how exactly these new independent variables are determined. Is there a method or is it a mixture of practice and luck?
Edit: since both responses (thank you) say it is mainly experience, lets extend the question.
Is a simplification via substitution of new variables always possible? For example, if we were to add diffusion terms to both equations
$$\frac{\partial V}{\partial t}+c\frac{\partial V}{\partial x}-D_V\frac{\partial^2 V}{\partial x^2}=\gamma(U-V)$$ $$\frac{\partial U}{\partial t}-D_U\frac{\partial^2 U}{\partial x^2}=\beta(V-U),$$ can we still make a useful substitution?
It's not just a guess. I'll use the symbol $\sim$ to indicate that two quantities have the same dimension. If any two things are added together they must have the same dimension. The left-hand side of the first equation implies that $$c\sim L/T$$ (length over time). Equating the left and right implies $$\gamma\sim T^{-1}.$$ Similarly, $$\beta\sim T^{-1}.$$ Since $x$ is a length, you have just two choices for its non-dimensionalized version: $$\xi=\frac\gamma cx\text{ or }\xi=\frac{\beta}{c}x.$$ Similarly, for the time variable, we can take either $$\tau=\gamma t\text{ or }\tau=\beta t.$$ In total, you have four choices for the non-dimensionalization. Which one you pick should depend on what you are doing. For instance, taking the first choices above for $\xi$ and $\tau$, the equation becomes \begin{align*}\partial_\tau V+\partial_\xi V=U-V\\\partial_\tau U=\frac{\beta}{\gamma}(V-U).\end{align*} Note that the time shift as you write it is not related to dimensions, it's about changing coordinates to simplify the linear operator. Also note that this is a different non-dimensionalization than you achieved before since it involves the dimensionless quantity $\beta/\gamma$. One or the other might be preferred depending on the application (if you are doing something that has an application...). For example perhaps you want to study the regime $\beta\ll\gamma$.
As a side issue, to eliminate the transport part of the equation, you could let $\eta=\xi-\tau$ to obtain \begin{align*}\partial_\tau V=U-V\\\partial_\tau U=\frac{\beta}{\gamma}(V-U).\end{align*} Again, this is not a matter of dimensions; it's based on the observation that $\partial_\tau+\partial_\xi=(1,1)^T\cdot\nabla_{\tau,\xi}$ so we've chosen $\eta$ to satisfy $$\nabla_{\tau,\xi}\eta\cdot(1,1)^T=0.$$ See any elementary treatment of transport equations for more details.
Edit: By adding more terms you've actually increased the number of possible non-dimensionalizations. Note that now $c/\gamma$, $c/\beta$, $(D_V/\gamma)^\frac12$, $(D_V/\beta)^\frac12$,... are all valid choices for the length scale.