So let's say I copy /var/lib/mysql and save it somewhere else, then uninstall/reinstall MySQL and put the original /var/lib/mysql back where it was. Will I have all of my databases without any trouble?
However, If you stop your MySQL server and copy everything under /var/lib/mysql, including ibdata1, ib_logfile0, ib_logfile1, and everything in subdirectories under /var/lib/mysql (*.frm, *.ibd files), and later on restore these files to some other server, using the same version of MySQL and same configuration (/etc/my.cnf or similar), then you'll be fine.
There are better ways to keep backup or transfer data between servers. See mysqlhotcopy, mysqldump, etc.