Best to export the data from the MySQL host database and import it into your MySQL database on your local computer. For clarification, the export file would contain a series of SQL insert statements, and optionally include drop and create table statements.
I'd install and use phpMyAdmin instead of the command line, on each.
The data files you refer to do exist in MySQL under the MySQL install directory 'mysql/data' but I wouldn't touch them, or assume that copying them to another server would 'just work'.