The other day I told you how to change the default search engine for Opera, so that the set engine will be the one used when you type in the address bar. You can also do the same with Firefox as well.
Similar to Opera, you can set your default search engine for the search box. For the address bar the search engine used is Google, using it's HTTPS search feature. If you wish not use Google and use something else, you'll have to get your hands dirty. You'll have to make a change in about:config, and then the change will be effective immediately.
In this tutorial I'll just change the search engine to Bing. You can use actually any type of search URL, but I'll talk about that in a bit.
First, open up Firefox and open a new tab. Click the URL bar, and type "about:config". If you get a warning, then go ahead and ignore it. This warning is there because we are actually changing the "raw" settings of Firefox and if you mess those up you can cause problems to the browser.
After you are at the about:config screen, type "keyword" in the search box. There should be two results, or more depending on what extra extensions you have installed. Locate the keyword.url setting, and right click and choose "modify". In the window that pops up, enter a search string. The format would be the full url of the search engine, all the way to the "?q". So for Bing, enter "http://www.bing.com/search?=". Everything after the "?q=" will be the search query, which is what you type into the Firefox address bar.
After you click OK, you should have the keyword.url setting in bold, showing that it has been manually changed (user set), which will override the defaults. To get rid of this setting (reset it), then right click on the setting and choose "reset".
The setting takes affect immediately. Mozilla does however want to warn you if you change to a search engine that uses an unencrypted connection, you loose a certain level of privacy. If you didn't know a few versions ago a Firefox update brought HTTPS Google searches by default.
If you have questions let me know in the comments below. If you want t visual way to do this I'll compose a video at your (the readers) request.