Resource for learning about the Laplacian on Riemannian manifolds
This (great) book of Peter Buser is about the spectrum of the Laplace Beltrami operator over manifolds. He also gives methods to build isospectral but not isometric manifold based on the Sunada Theorem (which answers the question: "Can one hear the shape of a drum?").
Here is the book entitled "The Laplacian on a Riemannian Manifold". But I did not study it in details. Instead, I would recommend Chapter III of Schoen and Yau's book "Lectures on Differential Geometry, which I have studied in details. It introduces many results about eigenvalues of Riemannian manifolds.