A good multi-variable calculus text recommended after completing Spivak's Calculus includes Wilfred Kaplan's "Advanced Calculus" and Piskunov's Volume 2. For more advanced study, Richard Courant and John Fritz's "Volume 2 Parts 1 and 2" are suggested. Additionally, Spivak's "Calculus on Manifolds" is a strong option, but prior knowledge of linear algebra is essential. Key linear algebra concepts to understand before tackling multi-variable calculus include linear combinations, matrix operations, inverse matrices, linear dependence and independence, determinants, linear transformations, and vector spaces.