People use measure theory in tandem with differential forms all the time—there's no contradiction whatsoever between the formalisms. Be aware, though, that the adjective “Riemannian” in the context of differential geometry refers to constructions depending on Riemannian metrics (which are “Riemannian” in the sense of originating in the work of Bernhard Riemann), not to Riemann integration.
Suppose that $M$ is a smooth $n$-manifold. By definition, it's locally diffeomorphic to $\mathbb{R}^n$, so that you can define a set $S \subset M$ to be measurable if and only if $x(S \cap U) \subset \mathbb{R}^n$ is Lebesgue measurable for every local coordinate chart $x: U \to x(U) \subset \mathbb{R}^n$. This gives you a $\sigma$-algebra of Lebesgue measurable sets on $M$ that correctly completes the Borel $\sigma$-algebra generated by the open sets on $M$ as a topological space. At this point, you have everything you need to define measurable functions, vector fields, differential forms, tensor fields, etc., in a manner compatible with calculations in local coordinates.
Now, suppose that $M$ is a Riemannian manifold, so that it comes equipped with a Riemannian metric $g$—again, the “Riemannian” here does not refer to Riemann integration, but to Riemann himself and his work on differential geometry. On any local coordinate chart $x : U \to x(U) \subset \mathbb{R}^n$, you can define a measure $\lambda_{g,x}$ on $U$ by setting
$$
\lambda_{g,x}(S \cap U) := \int_{x(S \cap U)} \sqrt{\det\left(g\left(\tfrac{\partial}{\partial x^i},\tfrac{\partial}{\partial x^j}\right)\right)} \,d\lambda
$$
for any Lebesgue measurable $S \subset M$, where $\lambda$ denotes Lebesgue measure on $\mathbb{R}^n$. By paracompactness of the manifold $M$, one can cover $M$ by a locally finite open cover of such local coordinate charts, and hence use a smooth partition of unity subordinate to this cover to patch these local scaled pullbacks of Lebesgue measure together into a single measure $\lambda_g$, the Riemannian measure [!] on $M$ with respect to $g$, which is a complete $\sigma$-finite measure on the $\sigma$-algebra of Lebesgue measurable sets in $M$.
Let me now describe the basic properties of $\lambda_g$.
The measure $\lambda_g$ is compatible with calculations in local coordinates, in the precise sense that $\lambda_g(S \cap U) = \lambda_{g,x}(S \cap U)$ for any Lebesgue measurable $S$ and any local coordinate chart $x : U \to x(U) \subset \mathbb{R}^n$.
If $g^\prime$ is any another Riemannian metric, then the Riemannian measures $\lambda_g$ and $\lambda_{g^\prime}$ will be mutually absolutely continuous $\sigma$-finite measures with smooth Radon–Nikodym derivative computable directly in terms of $g$ and $g^\prime$.
Suppose that $M$ is orientable, and let $\mathrm{vol}_g \in \Omega^n(M)$ be the Riemannian volume form defined by $g$. Then for any Riemann integrable $f$ on $M$,
$$
\int_M f \, \mathrm{vol}_g = \int_M f \,d\lambda_g,
$$
so that $\lambda_g$ really is the (completed) Radon measure on $M$ corresponding to the positive functional $C_c(M) \ni f \mapsto \int_M f \, \mathrm{vol}_g$ via the Riesz representation theorem. In other words, integration with respect to $\lambda_g$ really is the “Lebesgue-ification” of integration against the top-degree form $\mathrm{vol}_g$.
Once you've constructed the Riemannian measure on your Riemannian manifold $(M,g)$, the sky is now the limit—you can construct $L^p$ and Sobolev spaces of functions, vector fields, differential forms, tensor fields, etc., and in particular, you can use them to study, for instance, the geometric partial differential operators (e.g., generalisations of the Laplacian and the Dirac operator) and their associated partial differential equations (e.g., heat equations) to great mathematical effect. As a mathematical researcher, I'm personally most familiar with the mathematical ecosystem centred around the Atiyah–Singer index theorem, which relates quantities from algebraic topology to functional-analytic computations on Riemannian manifolds, but you should be aware, for instance, that Perelman's proof of the Poincaré conjecture involved the detailed analysis of a certain highly non-linear PDE for the Riemannian metric itself [!]. Perhaps the most accessible example of these methods in action is Hodge theory, which basically computes the cohomology of a compact manifold in terms of solutions of the Laplace equation (with respect to some Riemannian metric) on differential forms of various degrees.
P.S. People tend to take the extension of Lebesgue theory from $\mathbb{R}^n$ to manifolds more or less for granted, so precise accounts of this can be oddly hard to find. However, a precise if terse account of Lebesgue theory on manifolds can be found in Dieudonné's Treatise of Analysis, Volume 3, Section 16.22 (especially Theorem 16.22.2 and the following discussion). Dieudonné doesn't require a Riemannian metric, but the point is that Riemannian metric gives a canonical choice of Lebesgue measure in the sense of Dieudonné, in exactly the same way that it gives a canonical volume form in the orientable case. In fact, Lebesgue measures in the sense of Dieudonné can be identified with nowhere vanishing $1$-densities, and the construction of the Riemannian measure $\lambda_g$ is really the construction of the canonical $1$-density $\lvert \mathrm{vol}_g \rvert$ associated to $g$.
ADDENDUM
One can define a measurable $k$-form on $M$ to be a map $\omega : M \to \wedge^k T^\ast M$, such that the following hold.
- For every $m \in M$, $\omega(m) \in \wedge^k T^\ast M_m$ (i.e., $\omega$ is a set-theoretic section of $\wedge^k T^\ast M$).
- For every local coordinate chart $x : U \to x(U) \subset \mathbb{R}^n$, the pullback $(x^{-1})^\ast \omega : x(U) \to \wedge^k \mathbb{R}^n$ defined by
$$
(x^{-1})^\ast\omega := \sum_{i_1 < \cdots < i_k} \omega\left(\tfrac{\partial}{\partial x^{i_1}},\dotsc,\tfrac{\partial}{\partial x^{i_k}}\right) dx^{i_1} \wedge \cdots \wedge dx^{i_k}
$$
(with the usual abuses of notation) is measurable; this turns out to be equivalent to requiring that $\omega(X_1,\dotsc,X_k) : M \to \mathbb{R}$ be measurable (in the above sense) for any smooth vector fields $X_1,\dots,X_k \in \mathfrak{X}(M)$.
Now, suppose that $N$ is an oriented $k$-dimensional submanifold of $M$ (compact and without boundary, for simplicity), and let $x : U \to x(U) \subset \mathbb{R}^n$ be a local coordinate chart of $M$, such that $x(N \cap U) = V_{x,N} \times \{0\}$ for some open $V_{x,N} \subset \mathbb{R}^k$, and such that restriction of $x$ to a diffeomorphism $N \cap U \to V_{x,N}$ is orientation-preserving. Then we can define
$$
\int_{N \cap U} \omega := \int_{V_{x,N}} \omega\left(\tfrac{\partial}{\partial x^{1}},\dotsc,\tfrac{\partial}{\partial x^{k}}\right) d\lambda_{\mathbb{R}^k}
$$
whenever the Lebesgue integral on the right-hand side exists (with $\lambda_{\mathbb{R}^k}$ the Lebesgue measure on $\mathbb{R}^k$). We can then define $\omega$ to be integrable on $N$ whenever it's integrable in this way on $N \cap U$ for any suitable local coordinate chart $x : U \to \mathbb{R}^n$, and then, by exactly the same arguments as in the Riemann integral case, patch these local integrals into a global Lebesgue integral $\int_N \omega$, which turns out to be independent of all the choices of local coordinate chart and partition of unity made along the way.
