I'd suggest working with the vignette - that's the document I put together to essentially act as a user manual. The PDF you linked to is auto-generated as part of the submission to CRAN, and includes documentation for individual functions within the package. Some of these (i.e. call_mutations(), estimate_lineages(), and explore_mutations()) are likely to be useful to you, while there are others in there that can be thought of as "behind the scenes" functions - most users of the package would not have any reason to use these.
If you really want to get up and running quickly, at least in a very basic way, I'd probably suggest jumping to the "Example Analysis 1" section of the vignette. You should be able to just copy, paste, and run the code from Steps 1 and 2 of that section to start to analyze an example dataset. Then if you have your own VCF files (assuming they are SARS-CoV-2), you could just replace the system.file function and its associated options in Step 1 with a path to a directory on your computer where those VCF files are stored and rerun. You'll probably want to remove the name.sep option from that example code too, unless your files happen to be named in a similar pattern as the example files, where there is extra text after an '_' that can be trimmed off).
Good luck, and let me know if you have other questions or run in to any problems.