I'm seeing a bunch of questions such as "find all ring homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}/30\mathbb{Z}$," etc., lately, so I'm wondering about whether there's a general method for doing this between any two rings. What steps should I take? I'd appreciate an example in your answer, if you can manage one. Thanks so much!
EDIT: I should be clear here that I'd not necessarily like to see methods that work in every case, and I don't necessarily need a lot of rigour either. I'd just like to see what works for you in most practical cases.