The Church of St. Peter

How does the priest go about choosing our Sunday Scripture readings week by week?

The priest doesn't choose. The readings are already set out for every Sunday of the year in a 3-year cycle. This is called a "lectionary". We currently use the Revised Common Lectionary, and it provides four readings for each Sunday of the year.
The first reading always comes from the Old Testament, except in the Easter season, when it comes from the Book of Acts. The second reading is usually a psalm, although occasionally it is a "canticle" (a Biblical hymn). The next reading comes from one of the Letters of the New Testament or the Book of Revelation, and the fourth and final reading is always from one of the Gospels.
When we travel through the three year cycle of readings that comprise the Revised Common Lectionary (Years A, B and C), we read through much of the Bible, but not all of it. There are some verses from the Bible that are never read on Sunday mornings. One of the many purposes of the lectionary is to give us a sampling of the major themes and stories of the Bible. It also allows Christians (at least those who follow it) to read the same readings together week by week. This means that we at St. Peter's are reading and praying along with other Anglicans around the world, as well as many Roman Catholics, Presbyterians and Christians of other denominations. The new liturgical year begins on the First Sunday of Advent, so every Advent we begin a new cycle of readings.