Assume that $G$ is connected. Let $T$ be a maximal torus of $G$. Restriction induces a map $R(G) \to R(T)$. Note that $R(T)$ is a Laurent polynomial ring in $r$ variables where $r$ is the rank. Because the conjugates of $T$ fill up $G$, Peter-Weyl implies that this map is injective. Moreover, if $W$ is the Weyl group, then the image of this map lies in $R(T)^W$, and I think this map is always an isomorphism. So $R(G)$ is about as nice as possible: in particular, it is a finitely generated integral domain, so Noetherian of finite Krull dimension, etc. A lot is known about invariants of Weyl groups so you can probably get even more explicit information than this.

Assume that $G$ is connected. Let $T$ be a maximal torus of $G$. Restriction induces a map $R(G) \to R(T)$. Note that $R(T)$ is a Laurent polynomial ring in $r$ variables where $r$ is the rank. Because the conjugates of $T$ fill up $G$, Peter-Weyl implies that this map is injective. Moreover, if $W$ is the Weyl group, then the image of this map lies in $R(T)^W$, and I think this map is always an isomorphism. So $R(G)$ is about as nice as possible: in particular, it is a finitely generated integral domain, so Noetherian of finite Krull dimension, etc. A lot is known about invariants of Weyl groups so you can probably get even more explicit information than this.

Example. Let $G = \text{U}(n)$. Then $R(T)$ is a Laurent polynomial ring in $n$ variables, the Weyl group is $W \cong S_n$ acting by permutation, and the invariant subring is $\mathbb{Z}[e_1, ..., e_n, e_n^{-1}]$ where $e_i$ is the $i^{th}$ elementary symmetric polynomial, which corresponds to $\Lambda^i(\mathbb{C}^n)$.