This is a very basic question (most of the questions I ask are), but how and under what circumstances does a PC attain an increase in Ring or Skill? And are Ring and Skill increases of a PC limited to those determined by the starting stats of their school or family?

You can buy them at any time so long as you have XP. The point of curriculum skills is that they grant full value (1 XP spent <-> 1 XP toward next rank) towards school progression, as opposed to other XP expenditures that only count half towards school progression. Notably buying up rings always counts for half, so there's a tradeoff in bumping up skills/techniques for rank vs. buying up rings for dice pools. Check pages 97-98 if you want more detail on this.

Thank you.