There are actually a few different ways to do this, including modifying the functions.php file in your theme folder, or modifying the default-filters.php file in your wp-includes folder. But I like creating a separate php file and making it into a WordPress plugin, as described in this post by Katja Stokley. Create a text file called TurnOffSmartQuotes.php and put this in it.
Source Code for TurnOffSmartQuotes.php
Then just upload this file to your wp-content/plugins folder, login to your WordPress dashboard, select "Plugins," and activate your new plugin. You've just turned off smart quotes on your WordPress blog.