There are many more approaches than simple quantitative versus qualitative research, this is just one place to start. For an in-depth discussion of SoTL methodologies, see Miller-Young & Yeo (2015).
Next, you can start thinking about how you will collect data. Again, there are many possibilities. Some common methods include surveys, interviews, focus groups, observation, standardized tests, learning assessments, and experiments. It is most important that the data collection method matches the question and intended outcome.
For a detailed explanation of the process of designing a SoTL study, Elgie (2014) is a great place to start. The University of Calgary’s Taylor Institute for Teaching and Learning also has great resources for understanding the different components of SoTL research, like these short articles on the methodological considerations for SoTL research and planning a SoTL lesson study.
You can also find more resources for supporting research on the Library's Research Support page.