Relation between the homotopy classes of maps on a torus, and maps on a sphere

One case in which you can establish a simple relationship is when $Y$ is a loop space. Suppose that $Y\simeq \Omega Z=\mbox{map}_*(S^1, Z)$. Then there is a bijection $[{\mathbb T}^d, Y]_*\cong [\Sigma{\mathbb T}^d, Z]_*$. On the other hand, there is an equivalence $$\Sigma{\mathbb T}^d \simeq \bigvee_{i\ge 1} \bigvee_{d\choose i} S^{i+1}.$$

This follows, using induction, from the equivalence $\Sigma X\times Y\simeq \Sigma X\vee \Sigma Y \vee \Sigma X\wedge Y$ (Proposition 4I.1 in Hatcher's book). It follows that there is a bijection

$$[{\mathbb T}^d, Y]_*\cong [\bigvee_{1\le i\le d} \bigvee_{d\choose i} S^{i}, Y]_*\cong \prod_{i=1}^d\prod_{d\choose i} \pi_i(Y).$$

It is easy to extend this calculation to unpointed homotopy classes of maps, because $[{\mathbb T}^d, Y]\cong [{\mathbb T}^d_+, Y]_*$ and there is an equivalence $\Sigma {\mathbb T}^d_+\simeq \Sigma {\mathbb T}^d \vee S^1$.

Homogeneous spaces are usually not loop spaces, but for example topological groups are loop spaces, including the orthogonal and the unitary groups.

By the way, this answer on math.SE has a very nice description of the set $[{\mathbb T}^2, X]$ for a general space $X$. https://math.stackexchange.com/questions/36488/how-to-compute-homotopy-classes-of-maps-on-the-2-torus