I need to know the specific process for requesting permission to use
partial lyrics or the title of a song when writing a book or designing
a greeting card; should the songwriter or record company be used?

As to the title:  titles cannot be copyrighted, so you can use a song
title in your greeting card or book without copyright liability.  The
caveat is that a song title may also be a phrase in which someone
claims a trademark interest, so you would have to search and see if
the title is claimed as a trademark and then seek permission of the
trademark owner if needed.

As to the partial lyrics: generally the copyright to song lyrics is
owned by the author, i.e., the actual lyricist.  You can search the
copyright office records or you may wish to contact the record company
and inquire about the license since they might direct you to the
author or they may have been given the lyricist's authorization to
enter into licensing arrangements.