If we have some function $f(a,b,c,...)$, where $a,b,c,...$ can be stochastic or otherwise, then Ito's lemma is used to find $df(a,b,c,...)$.
1)
You can simply do raw Monte Carlo. Consider a contingent claim maturing in $6$ months. Then for each $i$-th simulation you can calculate:
$S(T)_i = S(t)e^{(r-q-\frac12 \sigma^2)0.5 + \sigma \sqrt{0.5}z_i)}$
where $z_i \sim N(0,1)$, and $\sqrt{\Delta t}(z_i)$ is equal in distribution to $W_{\Delta t}$.
2)
You can use the above methodology but create a sample path. For example you might want to generate a 6-point sample path. That is;
$S(t_i) = S(t_{i-1})e^{(r-q-\frac12 \sigma^2)\frac{0.5}{6} + \sigma \sqrt{\frac{0.5}{6}}z_i)}$
for $i \in \{2,3,4,5,6,7\}$.
3)
You can discretize the SDE itself using Euler-Marayama.
$\Delta S(t_i) = a(t,S)\Delta t + b(t,S)\Delta W(t_i)$
4)
You can discretize the SDE using Milstein:
$\displaystyle \ \ \Delta S(t_i) = a(t,S)\Delta t + b(t,S)\Delta W(t_i) + 0.5b(t,S)\frac{\partial b(t,S)}{\partial S}\bigg((\Delta W(t_i))^2 - \Delta t\bigg)$
5)
Consider the above methodologies to be the function $f(v)$, where $v$ are your random numbers. You can use a control variate $g(v)$ in the following fashion:
$\frac1N \sum_{i=1}^N [ f(v_i) - g(v_i) ] + E[g(v)]$.
In practice you want to use correlations and other stuff to improve the outcome; however this is how it's presented to students. $E[g(v)]$ might be the closed-form BS price, $g(v_i)$ might be the monte carlo BS price for a simple instrument, $f(v_i)$ might be some really complicated instrument that's closely correlated with $g(.)$.
6)
You can use antithetic sampling. That is,
$\text{MC estimate} = \frac12 [ f(v) + f(-v)]$.
There are some technical conditions you need to satisfy to make this worth the extra computation.