What Vim command(s) can be used to quote/unquote words?

surround.vim is going to be your easiest answer. If you are truly set against using it, here are some examples for what you can do. Not necessarily the most efficient, but that's why surround.vim was written.

• Quote a word, using single quotes
  ciw'Ctrl+r"'
  • ciw - Delete the word the cursor is on, and end up in insert mode.
  • ' - add the first quote.
  • Ctrl+r" - Insert the contents of the " register, aka the last yank/delete.
  • ' - add the closing quote.
    
    
• Unquote a word that's enclosed in single quotes
  di'hPl2x
  • di' - Delete the word enclosed by single quotes.
  • hP - Move the cursor left one place (on top of the opening quote) and put the just deleted text before the quote.
  • l - Move the cursor right one place (on top of the opening quote).
  • 2x - Delete the two quotes.
    
    
• Change single quotes to double quotes
  va':s/\%V'\%V/"/g
  • va' - Visually select the quoted word and the quotes.
  • :s/ - Start a replacement.
  • \%V'\%V - Only match single quotes that are within the visually selected region.
  • /"/g - Replace them all with double quotes.

Quote a word, using single quotes

ciw'Ctrl+r"'

It was easier for me to do it this way

ciw '' Esc P

Here's some mapping that could help:

:nnoremap <Leader>q" ciw""<Esc>P
:nnoremap <Leader>q' ciw''<Esc>P
:nnoremap <Leader>qd daW"=substitute(@@,"'\\\|\"","","g")<CR>P

If you haven't changed the mapleader variable, then activate the mapping with \q" \q' or \qd. They add double quote around the word under the cursor, single quote around the word under the cursor, delete any quotes around the word under the cursor respectively.