I'm doing 3rd year undergraduate geometry (an introductory subject), and we've been given formal definitions of terms like "$n$-manifold" and "smooth $n$-manifold." However, I tend to think about these concepts in simplified terms. My simplified definitions are:
An $n$-manifold is a subspace of $\mathbb{R}^k$ for some $k \in \mathbb{R}$ that is locally homeomorphic to $\mathbb{R}^n,$ regarded up to homeomorphism.
A smooth $n$-manifold is a subspace of $\mathbb{R}^k$ for some $k \in \mathbb{R}$ that is locally diffeomorphic to $\mathbb{R}^n,$ regarded up to diffeomorphism.
Question 0. Are these definitions basically correct? If not, why not?
Question 1. Can the concept of a Riemannian $n$-manifold be given a "simplistic" definition like those above that is basically correct?
Your intuitive descriptions are fairly ok for most manifolds, if you replace "subspace" by "subset" and if your formal definition on manifold includes paracompactness and second-countability. However, we do NOT identify homeomorphic topological manifolds or diffeomorphic smooth manifolds. A smooth manifold is defined in terms of a maximal atlas with smooth transition functions and two smooth manifolds are equivalent if they have compatible atlases. It is completely possible and frequent for two manifolds to NOT have compatible atlases (so have different smooth structures on them) but nevertheless be diffeomorphic. The standard example can be found here: https://mathoverflow.net/questions/33805/why-is-the-x-x1-3-atlas-on-r-diffeomorphic-with-the-x-x-atlas-on-r with detailed discussion.
Let me also note that although strong enough axioms guarantee that manifolds can be embedded into some finite-dimensional Euclidean space, it is actually better NOT to think of them as such. This is because manifolds can be embedded in many spaces in many incompatible ways; building your intuition around one such embedding can lead to stumbling blocks when you need to embed them in other spaces or in different ways.