`:set list`
This will show you whitespace characters like tabs and EoLs. It won't show spaces, however; to my knowledge that's not possible (except for non-breaking and trailing spaces), although in a monospace font any "space" that's not a tab would obviously be a space.
You can change the characters vim uses with the `listchars` option; type `:help listchars` to learn more about how to use that and what your options are.
This is what I use in my .vimrc file:
" Set some nice character listings, then activate list
execute 'set listchars+=tab:\ ' . nr2char(187)
execute 'set listchars+=eol:' . nr2char(183)
set list