There are many methods, and choosing one depends upon what type of research you are doing. Specifically how precise and valid does your findings have to be.
In qualititave research grounded theory and thematic analysis are two methods used in academia. For ux-studies I would recommend using an approach based on thematic analysis. This means that there are two stages analysis and synthesis.
This article on analysis and synthesis gives a good overview of the general steps:
1. Collect and organize the data. Make your data manageable.
2. Mine the data. Identify what you see.
3. Sort and cluster the data. Manipulate or reframe your data, as necessary.
4. Identify insights. Discuss, articulate, incubate, and socialize your insights.
There are tools, methods and strategies for doing this work. For instance using card sorting to sort comments into groups.
I suggest using excel, google sheets or airtable together with some ideation tool like realtimeboard. There are other tools that I haven't used like optimal workshops reframer, however it is quite expensive.
Its also interesting to read how professional ux researcher think and work, therefore I would suggest looking at polaris nuggets for inspiration.