Transfer Function of Electrical System
\$V_{in}\$ is switched between 0 and 13 Volts (i.e \$V_{in}(t) = 13u(t)\$)


You don't need to know this to find the transfer function. 

After you know the transfer function, you can multiply it by the Laplace-domain equivalent of the input voltage to get the output voltage.


  I am assuming that the first step is to apply a KVL anaylsis to the circuit.


That would work but it can also be done just using parallel and series combinations, along with the voltage divider formula, to get the transfer function. First step is to find the Laplace-domain representations for each circuit element.