Gröbner basis for finitely generated algebras

I am curious if there is a notion of how to find a Gröbner basis for any ideal $I$ of a finitely generated algebra $R\cong \mathbb{K}[x_1,\dots,x_k]/J$. I know that Gröbner bases are generaly developed as a tool for polynomial rings, but I wonder what fails in this case or in which cases it's possible to compute a Gröbner basis in $R$ as in a polynomial ring. Particularly, I wonder if given an algebraic variety $X$ and a subvariety $Y\subseteq X$ it's possible to find a Gröbner basis for $I(Y)$ if we know a basis for $I(X)$ or what fails when we try to do this.