I would say that it is not true that quotients of polynomials rings are "almost UFDs".

For starters, being a quotient of $k[x_1,\ldots,x_n]$ for some $n$ just says that the ring is finitely generated over $k$. If $k$ is a field these rings are reasonably nice but they can still be quite "badly behaved" and a long way away from having unique factorization.

For instance one can view the class group of a Dedekind domain $A$ as measuring how badly unique factorization fails in $A$. This group can be very large even when $A$ is finitely generated over a field - taking the ring corresponding to an elliptic curve with a point deleted gives examples with infinite class group (the class group is pretty much the underlying elliptic curve in this case).

In fact it is a theorem of Claborn that any abelian group occurs as the class group of some Dedekind domain. I am not sure how far one can get working with finitely generated algebras over a field, although there are other results in this direction that allow one to construct such examples by taking integral closures in quadratic extensions I think, or via rings of functions on elliptic curves (this second being work of Rosen originally).

And all of this is just in dimension 1.

I'm not so sure I understand the rest of the question - certainly at least in rings of integers one can compute (in principle) how the principal ideals generated by rational primes factor and one knows which primes ramify. Or do you mean in your specific example? I'm also not sure what sort of classification you are asking for and in what case?

I would say that it is not true that quotients of polynomials rings are "almost UFDs".

For starters, being a quotient of $k[x_1,\ldots,x_n]$ for some $n$ just says that the ring is finitely generated over $k$. If $k$ is a field these rings are reasonably nice but they can still be quite "badly behaved" and a long way away from having unique factorization.

For instance one can view the class group of a Dedekind domain $A$ as measuring how badly unique factorization fails in $A$. This group can be very large even when $A$ is finitely generated over a field - taking the ring corresponding to an elliptic curve with a point deleted gives examples with infinite class group (the class group is pretty much the underlying elliptic curve in this case).

In fact it is a theorem of Claborn that any abelian group occurs as the class group of some Dedekind domain. I am not sure how far one can get working with finitely generated algebras over a field, although there are other results in this direction that allow one to construct such examples by taking integral closures in quadratic extensions I think, or via rings of functions on elliptic curves (this second being work of Rosen originally).

And all of this is just in dimension 1.

I understand the other part of your question now and I like Ben's answer. In higher dimension trying to understand this will get worse - I think about the best general statement one can make about unique factorization is that every regular local ring is a UFD so if the geometric incarnation of your algebra is singularity free it is locally a UFD.