There are several methods in the Usability and User Experience (UX) research areas that have been used for capturing issues and experiences, for instance, surveys in a long-term use have been regarded as an effective way to get information about your users.
As a short answer, and talking about methodology, it depends on the users' profile. You can analyze and evaluate general customers, young users, elderly users, physicians and so on. Also, you need to have in mind if you want a qualitative measurement, a quantitative measurement or both. If you have no idea what options do you have, you can start here. Assuming that this is a general customer experience user survey, I would recommend both methods:
1) A method for quantitative measurement;
2) A method for qualitative measurement;
For (1) quantitative measurements you will need to have a perception about what are the user interface errors, where are your users clicking and where are your users taking more time on the user interface. On the other hand, measuring (2) qualitative characteristics of the user interface will put on what you ask here, you will need to ask simple questions to obtain engagement that you want.
You should use a tool called hotjar or a similar to it. With this tool, you can support the (1) quantitative measurement by using the heatmaps ("eyetracking") functionality and you can solve the (2) qualitative measurement by creating small and simple surveys.
I also recommend the use of Polls. Since it allows you to ask your site visitors an unlimited number of questions. You also can configure them to appear randomly.