I have attached a screenshot of the question I need help with. Which one would conduct electricity the best and why?

HF is a weak acid and C₂H₅OH is a covalent molecule. Covalent molecule does not dissociate in water so non conductor of electricity.
HF dissociates very less so conductivity is low.
Rest Pb(NO₃)₂ and KMnO₄ are strong electrolytes and dissociates completely.
Now we have to check which one gives more concentration of ions in the solution.
Pb(NO₃)₂ ---> Pb ²⁺ + 2NO₃^-
      0.10 M ------> 0.10  + 0.20 = 0.30 M ions concentrations

KMnO₄ (aq) -----> K⁺ + MnO₄^-(aq)

     0.10 M     ------>    0.10   + 0.10 = 0.20

More is the concentrations of ions ,more is the conductivity of the solution.
So  Pb(NO₃)₂  is the best conductor.