I would check out "Heat Kernels and Dirac Operators" by Berline, Getzler, and Verne. It covers quite a bit of territory:

-Characteristic classes: Much stronger than most books; develops Chern-Weil theory in the setting of principal bundles, includes the equivariant case

-Index theory: This is one of the standard textbooks for the heat kernel proof of the index theorem and local index theory in general

-Lie groups: Proves the Weyl and Kirillov character formulas

-Kahler manifolds & complex geometry: Proves the Riemann-Roch formula as a special case of the index theorem, but otherwise not much

-Not much symplectic/poisson geometry (though maybe a little in the discussion of coadjoint orbits)

-Riemannian geometry: Proves Gauss-Bonnet-Chern and does some serious computations with curvature, but no comparison theorems

-Geometric analysis: heat kernels and Dirac operators are after all the theme of the book, but there's not really much discussion of standard elliptic operator theory or pseudo-differential operator theory, and there are no nonlinear operators

For the areas where the coverage is poorer - Riemannian geometry, complex manifolds / algebraic geometry, symplectic / poisson geometry, non-linear geometric analysis - a more focused book is probably required because the techniques are much more specialized. For Riemannian geometry you want the comparison theorems and discussion of non-smooth spaces (e.g. Burago-Burago-Ivanov is great). For complex manifolds you want a discussion of sheaf cohomology and Hodge theory (probably Griffiths and Harris is best, but I like Wells' book as well). For symplectic manifolds you want some discussion of symplectic capacities and the non-squeezing theorem (I think McDuff and Salamon is still the best here, but I'm not sure).

I would check out "Heat Kernels and Dirac Operators" by Berline, Getzler, and Verne. It covers quite a bit of territory:

-Characteristic classes: Much stronger than most books; develops Chern-Weil theory in the setting of principal bundles, includes the equivariant case

-Index theory: This is one of the standard textbooks for the heat kernel proof of the index theorem and local index theory in general

-Lie groups: Proves the Weyl and Kirillov character formulas

-Kahler manifolds & complex geometry: Proves the Riemann-Roch formula as a special case of the index theorem, but otherwise not much

-Not much symplectic/poisson geometry (though maybe a little in the discussion of coadjoint orbits)

-Riemannian geometry: Proves Gauss-Bonnet-Chern and does some serious computations with curvature, but no comparison theorems

-Geometric analysis: heat kernels and Dirac operators are after all the theme of the book, but there's not really much discussion of standard elliptic operator theory or pseudo-differential operator theory, and there are no nonlinear operators

For the areas where the coverage is poorer - Riemannian geometry, complex manifolds / algebraic geometry, symplectic / Poisson geometry, non-linear geometric analysis - a more focused book is probably required because the techniques are much more specialized. For Riemannian geometry you want the comparison theorems and discussion of non-smooth spaces (e.g. Burago-Burago-Ivanov is great). For complex manifolds you want a discussion of sheaf cohomology and Hodge theory (probably Griffiths and Harris is best, but I like Wells' book as well). For symplectic manifolds you want some discussion of symplectic capacities and the non-squeezing theorem (I think McDuff and Salamon is still the best here, but I'm not sure).