I'm learning this stuff myself so take this with a large grain of salt but a commenter on this question suggested Warner, Foundations of Differentiable Manifolds and Lie Groups.
At a glance it looks like it goes through some of the usual topics but then does the de Rahm theorem using sheaves, so you might get along with it. Apart from Ch.5, though, I'm not sure how different it is from a standard treatment. It's a GTM book with minimal prereqs, and if you already know about sheaves it's probably a fairly gentle read.
I'd be interested to know how those in the know regard this text in relation to (what I take to be) the more usual textbooks.
You shouldn't expect too many direct connections to Lebesgue integration. On smooth manifolds we want everything to be nicely differentiable, so there isn't much need to deal with exceptional sets of measure zero or anything like that, at least at the introductory level. Measure theory is certainly relevant once things like Haar measure get introduced, but that is far and away not part of an introductory sequence.
Differential forms generalize the line integral rather than the Lebesgue integral. In fact Lebesgue integrals are meant to deal with an entirely different problem (namely convergence of integral sequences) than the line integral or differential forms. The point of differential forms is to allow a definition of integration on manifolds that does not rely on the choice of parametrization of the manifold. The same manifold can be parametrized using many sets of coordinates (think $\mathbb{R}^3$ with cartesian, cylindrical, and polar coordinates), and for any quantity or concept to be well-defined on manifolds we must demonstrate that it does not depend on this choice of parametrization. Differential forms are therefore crucial to defining integration on manifolds; in addition, they are important in understanding an algebraic structure called the de Rham cohomology of a smooth manifold, which encodes valuable topological information.
Differential geometry in dimensions 2 and 3 falls into what we call the theory of surfaces and curves. Within this subject are many spectacular results that introduce the student to concepts that are important in the study of arbitrary dimensions. Among these should be (in a good course, anyway) the distinction between intrinsic and extrinsic geometry, the role of the metric, the role of curvature, the idea of the differential and the tangent space, geodesics, and expanded notions of differentiation (e.g. covariant). Being familiar with these, the student is prepared to study how the concepts generalize (or fail to) in arbitrary dimensions. Another advantage is that curves and surfaces are not only easier to visualize, extrinsic geometry (geometry that can depend on parametrization) has a greater role to play in lower dimensions, so the student gets his hands dirty doing actual calculations and studying examples that can be pictured clearly. It is impossible to do higher geometry if you lack these two capacities.
It seems evident to me that you'd much prefer the more general approach of Riemannian geometry and working with manifolds of arbitrary dimension. I advise against it if you haven't seen 2 and 3 dimensions yet (and if you haven't got a whole lot of linear algebra under your belt). By dismissing curves and surfaces you'll be missing out on a lot of important geometry specific to these dimensions, plus a lot of practice and intuition-building. Definitions, even analogous ones, in Riemannian geometry are not nearly so intuitive as they are in 2 and 3 dimensions. Depending on the text, they can be extremely abstract; in particular, the notion of a tangent space becomes significantly more subtle than in $\mathbb{R}^3$. Of course, if you thrive on abstraction, then by all means try reading Riemannian geometry instead, but be warned: the geometer that can't compute is no geometer at all, no matter what complicated theorems and definitions are floating around in his noggin.
You may also want to study some algebraic topology, especially homology and cohomology, and some differential topology; this is where differential forms really start to shine.
As for what chapters to study: all of them are relevant at some point, but pay particular attention to regular surfaces, differentials, tangent spaces, the first and second fundamental forms, Christoffel symbols, Gauss curvature, isometries and conformal maps, parallel transport, and geodesics. (This is ~75% of a typical one-semester curves and surfaces course. Yes, basically everything generalizes to higher dimensions.)
Best Answer
I stumbled upon a good series on Riemannian Geometry, https://www.youtube.com/watch?v=hh-q1q4bmBo&list=PLbMVogVj5nJSm4256vuITlsovUT1xVkUL&index=1
Any more suggestions are welcome