How to calculate concentrations in a diluted mixture of multiple solutions?
Am I somehow supposed to use $M_1V_1=M_2V_2$?

Yep!
The problem has given you three different solutions:

Iron(III) nitrate
Potassium thiocyanate
Nitric acid

You know the concentrations of each, and you know (or can calculate) the volumes of each that are mixed to yield the final solution.  What you'll actually be using in the calculation are three different equations:

$M_{1i}V_1 = M_{1f}V_f$
$M_{2i}V_2 = M_{2f}V_f$
$M_{3i}V_3 = M_{3f}V_f$

$M_{\#i}$ is the initial concentration of each of the three chemicals in their separate starting solutions, and $M_{\#f}$ is the final concentration of each after mixing, that the problem is asking for (except for the $\ce{HNO3}$).