You didn't mention which Linux Mint version you were using, and yes that is relevant to answer your question. Older versions of Linux Mint include the Linux Mint Google CSE on Firefox, but at least since Linux Mint 13 this hasn't been included any longer. That's been changed because of user feedback on that
Linux Mint 13 and 14 include Yahoo! as the default search engine for the search bar, and Duck Duck Go as default search engine for the awesome bar. To change the awesome bar, indeed you'll have to edit the keyword.URL in about:config as Firefox doesn't offer any normal way to edit that AFAIK.
To add Google to your search bar, you have two options. First, as detailed by Orbmiser above, click the down arrow icon in your search bar, select "Manage Search Engines", select "Get more search engines", click on the Google icon under "Commercial Search Engines", then click on the down arrow icon in your search bar again and select "Add Google". On Linux Mint 13 and 14, this gives you the regular Google search engine.
The second option you have, and unfortunately Firefox hasn't integrated this in its browser somehow, is the Mozilla Mycroft Project: http://mycroft.mozdev.org/
. You can find many, many, search engine plugins there, which you can install easily with one click.
Go to http://www.google.com
Right click in the search field and select 'Add to Search Bar'.
Then go to the search bar and move it up the list.
This doesn't work unless you have installed the "Add to Search Bar" add-on to Firefox. Note that this add-on gives you a more limited experience than using the other methods, as for example search suggestions don't work this way.