Deriving the Euler Equation
The question is quite straightforward, and you do not need the first
step that you have. You have (for some reason) a different multiplier
here for each time period. That is not the case- you simply have a
no-ponzi scheme condition, aka a transversality constraint. The problem
is expressed as $max\,\sum\beta^{t}U(c_{t})$ st $c_{t}+k_{t+1}\leq f(k_{t})$.
The stream of utility to the agent is $U=\sum\beta^{t}U(c_{t})=U(c_{0})+....\beta^{t}U(c_{t})+\beta^{t+1}U(c_{t+1})+...+\beta^{T}U(c_{T}).$

Substitute in $f(k_{t})-k_{t+1}$ for $c_{t}$. Then, we derive wrt
$k_{t+1}$ and set equal to 0:$\frac{\partial U}{\partial k_{t+1}}=\beta^{t}U'(c_{t})-\beta^{t+1}U'(c_{t+1})f'(k_{t+1})=0\Rightarrow U'(c_{t})=\beta U'(c_{t+1})f'(k_{t+1}).$

You do not need the other equation you referred to.