I think the easiest way is to use a plugin.
A nice simple plugin to do as you describe can be found at http://ww.wp.xz.cn/extend/plugins/social-media-widget/, I’ve used it on a couple of sites in the past.
If you’ve never installed a plugin before, download it and follow these steps;
- Upload the social-media-widget folder and all contents to /wp-content/plugins
- Activate the plugin through the ‘Plugins’ menu in WordPress
- Go to Appearance->Widgets and activate the widget (Social Widget), set options and click save
If your theme is widgetised, you can just drag the Widget into ‘Footer’ on the Appearance->Widgets page.
Otherwise just Google how to Widgetize your footer.
Although it’s a pain I think the best way forward is to install WordPress again in the correct directory.
Here is a guide if you want to keep the /wordpress directory though;
http://codex.ww.wp.xz.cn/Giving_WordPress_Its_Own_Directory
