I know we can use Lagrange's Theorem for groups, but can we also use them for rings? So, suppose we have $R$ as a ring and $I$ as an ideal. Then would it be true to say $R$ only has $|R|/|I|$ distinct cosets? What are the limitations?
Yes, this works perfectly fine. A ring is in particular a group with respect to the addition operation, and any ideal is in particular a subgroup. So any general statement about groups and subgroups will also be true for rings and ideals.
(Note that it doesn't really make sense to say there are $|R|/|I|$ distinct cosets when these sets could be infinite, division of infinite cardinalities cannot always be uniquely defined. But what is true always is that $|R|=|I|\cdot|R/I|$. This doesn't have anything to do with rings in particular though; this is just the correct way to state Lagrange's theorem for infinite groups.)