How do you track mentions about your company and relevant keywords on forums and blog posts, etc.? Are there any free tools to use?

Then you can use URL tracking codes to track links and view them in Google Analytics which are both free.

You can use advanced Google Search Operators to view relevant keywords in forums. An example would be like this one.

cars site:someforum.com

This typed into the search engine above will show you all of the pages that mention cars in only the website someforum.com.