Suppose $X$ and $Y$ are algebraic varieties, where $X$ is a right $G$-set, and $Y$ is a left $G$-set. Then $X\times_G Y$ is usually defined be the quotient of $X\times Y$ by the diagonal left action where $g$ acts by $(g^{-1},g)$.
Is it roughly correct to informally picture elements of this quotient as pairs $(x,y)$, where we can "commute" elements of $g$ across the comma, so that $(x,gy)=(xg,y)$ in $X\times_G Y$? I'm thinking of this as being analogous to the tensor product $A\otimes_R B$ where we view simple tensors concretely as satisfying relations like $a\otimes rb=ar\otimes b$, if $A$ is a right $R$-module, and $B$ a left $R$-module.
If I am incorrect, what is the best way to think of this quotient when attempting to do something hands on, like exhibiting an explicit isomorphism of $G$-sets or something like that?
Taking quotients of algebraic varieties by group actions is generally poorly behaved - the general goal of geometric invariant theory is to understand / work around this.
So, at this level of generality, the description you give of the balanced product $X \times_G Y$ is not correct. If you were only taking a quotient as sets then yes, $X \times_G Y$ would just be the orbit space of $X \times Y$ as a $G$-set.
But that quotient may be topologically ugly (e.g. non-Hausdorff), and may not exist as an algebraic variety. (And conversely, a different variety $Z$ may exist that satisfies a universal property with respect to G-equivariant morphisms from $X \times Y$, hence acts as a 'categorical' quotient. Typically the points of $Z$ correspond to closures of orbits rather than individual orbits, identifying any two orbits whose closures intersect.)
For example, let $Y$ be a point with a trivial $G$-action, so that $X \times_G Y = X/G$. Then for instance $X = \mathbb{A}^1$ with the scaling action by $G = \mathbb{G}_m$ ($t \cdot x \mapsto tx$) results in a poorly-behaved set quotient: two points, one of which is in the closure of the other. By contrast, the closest algebraic variety is just one point, corresponding to the closure of the large orbit (and is just $\mathrm{Spec}$ of the invariant subring $k[x]^G \subset k[x]$).