I'm not sure about a major-mode that already does this, but you can make one easily enough using define-derived-mode
(define-derived-mode rich-text-mode text-mode "Rich Text"
"text mode with string highlighting."
;;register keywords
(setq rich-text-font-lock-keywords
'(("\"\\(\\(?:.\\|\n\\)*?[^\\]\\)\"" 0 font-lock-string-face)))
(setq font-lock-defaults rich-text-font-lock-keywords)
(font-lock-mode 1))
Alternatively, you can add a hook to text-mode:
(defun add-quotes-to-font-lock-keywords ()
(font-lock-add-keywords nil '(("\"\\(\\(?:.\\|\n\\)*?[^\\]\\)\"" 0 font-lock-string-face))))
(add-hook 'text-mode-hook 'add-quotes-to-font-lock-keywords)
Generally speaking, a good mode for editing any text is org-mode. It does not font-lock strings by default, though.