As a technically lay person who does care about this,
what are some tools that I can use to write web copy
with typographically correct apostrophes and quotes
I use Textpattern CMS, which has built-in support for Textile (they were built by the same developer.) Textile converts marked-up text input to valid, well-formed XHTML and also inserts character entity references for apostrophes, opening and closing single and double quotation marks, ellipses and em dashes.
http://textpattern.com/