Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
They should feel similar! In fact, every ring is a group, and every field is a ring. A ring is a group with an additional operation, where the second operation is associative and the distributive properties make the two operations “compatible”.
A field is a ring such that the second operation also satisfies all the group properties (after throwing out the additive identity); i.e. it has multiplicative inverses, multiplicative identity, and is commutative.