Path for learning linear algebra and multivariable calculus

Try Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds by Theodore Shifrin. It shows how the two subjects are related.

If you want to start with abstract linear algebra, try Axler's Linear Algebra Done Right.

I don't think Apostol has enough for a first course in Linear Algebra, why don't you try: Linear Algebra by Hoffman and Kunze. The linear algebra bits in Artin's Algebra is also very good.

For multivariable calculus, I found Munkres' Analysis on Manifolds much easier to digest than Spivak.

If you want something requiring less mathematical maturity then you could try: Hubbard, Vector Calculus, Linear Algebra and Differential Forms. This is a fantastic text.