I have neglected to pay attention to the vertical lines on the staff. I know they indicate when to stop, repeat, etc., but I don't know the particulars. I searched the books I have and couldn't find the answer.
The lines I think you are talking about are called barlines. This image summarises them pretty well:
A single vertical line going across the staff marks the end of a bar. A bar consists of a specific number of beats, as designated by the time signature of the piece. You can read more about time signatures, measures, basic barlines and more here.
A double vertical line going across the staff marks the end of a section, or sometimes a phrase. They also commonly appear before important changes, such as a key change, time signature change, tempo changes or other stylistic changes. From another question on this site, here is how double bar lines can be used to separate changes in key signatures:
I think that a double bar line doesn't always indicate the end of a bar though (citation needed). For example if you change key in the middle of a bar.
A repeating section is designated by a thin and thick vertical line, with two dots. If the thicker line appears first, and the dots last, it is the start repeat. If the dots appear first, then the thinner line, then the thicker one, it is the end repeat.
Sometimes, starting repeat barlines might not appear, and an end repeat barline will appear at the end of the piece - this means repeat the entire piece. When you reach the end repeat sign, you go to the start repeat sign, or the start of the piece if there isn't one.
Finally, the end of the piece is usually notated with a thin vertical line, and a thicker one. The final barline normally won't have two dots next to it, unlike a repeating one.