Stackelberg with 3 firms
Start with the second stage, this is just Cournot competition between firm 2 and firm 3. You can solve this for the Nash equilibrium by setting the first order condition for firm 2 and firm 3 and solving these two equations, taking $q_1$ as given. This will give you quantities $q_2$ and $q_3$ in terms of $q_1$ which you can then plug into the profit function of firm 1 and you can maximize (i.e. find which $q_1$ firm 1 should choose to make sure the Nash equilibrium in stage 2 will be the most favourable Nash equilibrium possible for firm 1). So, step by step:


Start with the second round, find the Nash equilibrium by solving the following two equations:
\
$$\pi_2'(q_1,q_2,q_3)=0\\
\pi_3'(q_1, q_2, q_3)=0$$
Which will give you:
$$q_2=q_3=f(q_1)$$
plug this into the profit function of firm 1 and maximize this expression which has $q_1$ as choice variable.