Overlapping Generations model: Social Planner solution
Start by writing the Lagrangian (as it sounds like you have already done):

$$L=\sum_{t=0}^{\infty} \{\beta^t(\ln C_t^y + \ln C_t^o) + \lambda_t (C_t^y +\frac{C_t^o}{1+n} +k_{t+1}(n+1)-k_t-f(k_t))\}.$$

First order conditions are given by 
$$\frac{\partial L}{\partial C_t^y} = \frac{\beta^t}{C_t^y} +\lambda_t = 0$$
$$\frac{\partial L}{\partial C_t^o} = \frac{\beta^t}{C_t^o} +\frac{\lambda_t}{1+n} = 0$$
$$\frac{\partial L}{\partial k_{t+1}} = \lambda_{t-1}(1+n) -  \lambda_t(1 + f'(k_t)) = 0$$
$$\frac{\partial L}{\partial \lambda_{t}} = C_t^y +\frac{C_t^o}{1+n} +k_{t+1}(n+1)-k_t-f(k_t)=0.$$

First notice the first two equations give us 
$$\frac{C_t^y}{C_t^o} = 1+n.$$
Also notice that the first equation implies $\lambda_t = -\frac{\beta^t}{C_t^y}$. Substituting this into our third equation gives 
$$\frac{\beta^{t+1}}{C_{t+1}^y}(1+n) = \frac{\beta^{t}}{C_{t}^y}(1+n)(1 + f'(k_t)$$
or equivalently
$$\frac{\beta C_t^n}{C_{t+1}^y} = (1 + f'(k_t)).$$

Finally, we can write our system of four equations with three equations, having removed $\lambda$ from the system:
$$\frac{C_t^y}{C_t^o} = 1+n$$
$$\frac{\beta C_t^y}{C_{t+1}^y} = (1 + f'(k_t)).$$
$$C_t^y +\frac{C_t^o}{1+n} +k_{t+1}(n+1)-k_t-f(k_t)=0.$$

Dropping time subscripts, i.e. applying the steady-state condition, gives
$$\frac{C^y}{C^o} = 1+n$$
$$\beta = (1 + f'(k)).$$
$$C^y +\frac{C^o}{1+n} +nk-f(k)=0.$$

Hence, you are left with a system of three equations with three unknowns $C^y$, $C^o$, and $k$. Solve the system to arrived at a steady-state solution.