I'm asking this as a non-RC: How does a priest get to a diocese in the first place? Does the diocese offer the priest the position? Or is the priest assigned to the diocese by someone else?
Yes. Let us suppose, for the sake of discussion, that you were a Catholic man living in West Virginia ... and you thought perhaps that you were called to the priestly ministry. The Diocesan website offers you some guidance on how to go about pursuing that possible vocation. Note that Step 2 "The Application Process" involves an interview with the Bishop.
To add on to what arrogantbustard said, some dioceses have their own seminaries and the students from that diocese tend to stay in those dioceses.
After they are ordained they are assigned to a parish and then get reassigned every so often to another parish.
It usually happens in one of three ways:
1) A man becomes a priest in the diocese he lives in. He discerns that he is called to the priesthood, applies to the seminary for the diocese he lives in, attends seminary for 7 years, is eventually ordained, and works in that diocese for the rest of his life.
2) A man decides to become a priest in a diocese other than his own - for whatever reason. He might like the idea of working for a particular bishop more than the one in his diocese, etc.
3) A man discerns a vocation to the priesthood and joins a particular order: Jesuits, Franciscans, Dominicans, or any other, and goes (sometimes) very far from home to train for that religious order.