http://www.ehow.com/how_11385212_troubleshoot-reset-connection-firefox.html

https://support.mozilla.org/en-US/kb/server-not-found-connection-problem

https://support.mozilla.org/en-US/kb/firefox-cant-load-websites-other-browsers-can

https://support.mozilla.org/en-US/kb/secure-connection-failed-error-message

https://support.mozilla.org/en-US/kb/connection-untrusted-error-message

http://kb.mozillazine.org/Error_loading_websites

This Connection is Untrusted is sometimes caused because the computer system clock is wrong. Check the time / date / time zone settings.

Since you use Avast, one possible reason for this is its Web Shield feature that intercepts your browsing requests and filters the results. In order to filter secure sites, Avast must present Firefox with a "fake" certificate for the sites. Naturally, Firefox rejects the fake certs.

Have you created any "exceptions" in response to secure certificate errors? If so, when you're on one of those sites, click the padlock (or gray warning triangle) in the address bar, then More Information, then View Certificate, and check the "Issued by" section. If it shows Avast, then getting Firefox set up to work with Avast is the next step. If it shows something else, do you see any pattern when you check a few sites?