Start by deformation-retracting $G$ to its maximal compact, so that $G/T$ will be a flag manifold with a Bruhat decomposition, obtainable by Morse theory as in Agol's answer.

If you want the $\pi_2$ anyway, consider the long exact sequence on
homotopy of $T \to G \to G/T$, giving
$$ \ldots \to \pi_2(T) \to \pi_2(G) \to \pi_2(G/T) \to \pi_1(T) 
\to \pi_1(G) \to \ldots 
$$
The first obviously vanishes, and the last does too by assumption.
We know $\pi_2(G/T)$ is free of dimension $rank(G)$ by the
Bruhat decomposition, so the surjection $\pi_2(G/T) \to \pi_1(T)$
is between free abelian groups of the same finite rank. Hence it's
also injective. So its kernel $\pi_2(G)$ is zero.