I think that a quality department should develop three areas of expertise:
For quality control you can start with the risk-based thinking and developing a quality control plan from scratch: what to control, where, with what frequency, how, by whom, where to record, whit what specifications.
For quality improvement you start with the facts collected with quality control and customer satisfaction using tools to find trends, find priorities to improve your system.
Root cause analysis is fundamental for quality improvement and can be one of the bases for developing knowledge about how to design quality into products and services from the beginning.
The following material will provide you more information:
Free webinar on demand - Measurement, analysis, and improvement according to ISO 9001:2015 – https://advisera.com/9001academy/webinar/measurement-analysis-and-improvement-according-to-iso-9001-2015-free-webinar/-2015-free-webinar/