As previous questions about books on representation theory and Lie theory indicate, there are a lot of them out there aimed at different parts of the subject. (So maybe community-wiki is indicated?) It's good to be clear at the outset that the problem of finite dimensional tensor product decomposition over $\mathbb{C}$ is essentially the same for general linear groups and for their Lie algebras. However, the group approach is sometimes more flexible.

The "classical" Schur-Weyl approach is exposed in many books, including classical texts by Boerner and others aimed at physicists. One mathematical source I'm fond of is the straightforward chapter in the Springer GTM 225 book Symmetry, representations, and invariants by Goodman and Wallach. Also, the symmetric group background needed is well covered in the first part of the earlier Springer GTM 129 Representation Theory by Fulton and Harris; but it may be alittle harder to extract from their book a focused account of Schur-Weyl duality.

The more "modern" thinking of Littelmann and others (which is combinatorial and aims at avoiding formulas with many cancellations) may not yet be as readily accessible in book form.

There are also some Russian alternatives which others will surely want to advocate for.

Once you get into the decomposition of arbitrary tensor products, rather than just tensor powers of the natural representation, life does get more complicated and you can't expect miracles from the textbook sources.