My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd.
My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n \times S^k$ is parallelizable. In particular, there exists a global non-vanishing vector field, say $X$. It follows from Poincare-Hopf index theorem that $$\chi (S^n \times S^k)=0$$
On the other hand, by Kunneth formula, $H_*(S^n \times S^k)=H_*(S^n)\otimes H_*(S^k)$; and it follows that $$\chi (S^n \times S^k)=1+(-1)^n+(-1)^k+(-1)^{n+k}=(1+(-1)^n)(1+(-1)^k)$$ Hence, a necessary condition should be "$n$ or $k$ is odd".
Question (1) Is the condition "$n$ or $k$ is odd" a sufficient condition too?
(2) Are there some systematical methods to solve this kind of problem---determing tangent bundle or even fibre bundles over a specific manifold?
(3)Concerning the reference books related to "bundle theory", I only know Hatcher's book "Vector bndles and K-theory" and Bott's book "Differential forms in algebraic topology". , are there any other more comprehensive reference books else?
Please help! Thank you very much!
(1) Yes. Use the fact that the $TS^n$ are stably trivial. (Explicitly, embed $S^n$ in $\mathbb{R}^{n+1}$ and note that the normal bundle is trivial, since it clearly admits a nowhere-vanishing section. Now note that $T(S^n\times S^m) = TS^n \oplus TS^m$ and induct on $n, m$.)
(2) There are certainly a lot of results about classifying bundles by characteristic classes, up to stable trivialization, for specific classes of manifolds (e.g., closed, oriented $3$-manifolds), etc., but I don't know of a convenient algorithm.
(3) Hatcher's and Bott's books are exceptional, but I like Husemoller's "Fiber Bundles".