I want to track what people/visitors are searching on in that Google CSE. Is that possible and how?
Yes, it is possible using Site Search reports in Google Analytics:
- The Site Search reports in Google Analytics provide data and metrics on how people use search on your site. If you already have a
Google Analytics account (using the same Google Account you use for
your search engine), we recommend that you configure these reports
directly from the Custom Search control panel. The Site Search reports
provide the following data:
- Search usage: What did users search for on your site? Did they refine their searches? Where did they go after searching?
- Trending terms: What are people searching for?
- Pages: Which pages on your site have the most search usage?
Follow the instructions here to Configure Google Analytics Site Search. For the "query parameters" in step number 6 there, see the last step here: Install Google Analytics Site Search In 5 Steps
Alternatively, using a front-end search form as covered in this question, you could develop custom tracking suited to your specific needs.