A plain .tar archive created with cf (with or without v) is uncompressed; to get a .tar.gz or .tgz archive, compress it:
gzip < my_files.tar > my_files.tgz You might want to add -9 for better compression:
gzip -9 < my_files.tar > my_files.tgz Both variants will leave both archives around; you can use
gzip -9 my_files.tar instead, which will produce my_files.tar.gz and delete my_files.tar (if everything goes well). You can then rename my_files.tar.gz to my_files.tgz if you wish.
With many tar implementations you can extract archives without specifying the z option, and tar will figure out what to do — so you can use the same command with compressed and uncompressed archives.