What does Pb(NO₃)₂ + KI form?
This is just a metathesis (double replacement) reaction.

We know that nitrates are soluble, and alkali metals are soluble (at least in these general cases).

Let's ignore balancing for now, and do that last.

$\ce{Pb(NO3)2 + KI -&gt; Pb^{+2} + NO3^- + K+ + I-}$

$\ce{Pb^{+2} + NO3- + K+ + I- -&gt; PbI2 + KNO3}$

But the $\ce{KNO3}$ is also soluble, and would dissociate.

$\ce{Pb^{+2} + NO3- + K+ + I- -&gt; PbI2 + K+ + NO3-}$

Evidently, we can cancel out the spectator ions $\ce{K+}$ and $\ce{NO3-}$ and are left with a net ionic equation (which is easily balanced).

$\ce{Pb^{+2} + 2I- -&gt; PbI2}$