I like Wilfred Kaplan Advanced Calculus to start, also Piskunov Volume 2 is pretty decent. For more advanced, you can see Volume 2 Part 1 and Part 2 of Richard Courant and John Fritz.
You could move on to calculus on manifolds by spivak, but you would need some linear algebra first (which you should probably learn before multivariable calculus anyways) . Specifically linear combinations, matrix operations and inverse matrices, linear dependence/independence, determinants, linear transformations, spaces and basis for spaces would all be helpful to know.