For the Address Bar

1. Type about:config in you search bar as below

   

2. You'll then have a humourous notification box come up in the middle of the browser telling you the following:

   What you'll be agreeing to is that you'll be able to modify values of certain Firefox variables which change the overall user experience (so I strongly recommend you don't modify anything you don't need/know about).

3. Once you've clicked on "I'll be careful, I promise" the following search bar and list will come up:

   

4. Within the searchbar look up keyword and you should have the entries in the screenshot below

   

5. Lastly right click on keyword.url and click Modify and change it to whatever search engine you want, the example I've given here is DuckDuckGo (replace DuckDuckGo with any other search engine of your choice):

   

For those interested/wondering the ?q= at the end of the URL you post (as far as I know) is just how search engines determine the query required to be sent to the search engine.

For the search bar on the right

Just click on the triangle to the left and a drop down list will give you a choice of what search engine to use.

If your search engine isn't already there you will need to add it manually by clicking on the Manage Search Engines option at the bottom of that list of search engines and it will open up the following window:

Once there you simply click the Get more search engines... URL in orange and you can choose from a wide list which sites to add to your list so you can have them in your right toolbar at the ready.

Hope this answers your question.

Edit: I recommend users have a look at Dan's answer below, not exactly what was asked but still a very useful feature, well worth a read.

I will try to explain it...
the command sh -c 'firefox "https://www.google.com/search?q=$(xclip -o)"' will open firefox in www.google.com searching the words you have selected with your mouse.
To use it , you should install "xclip"

1. Go to the terminal and type sudo apt-get install xclip
2. Then you can create a script with the command, create a "new document" and named as "search-google.sh"
3. Open the file with gedit and put these lines:
4. Save the file in your home eg: /home/user/search-google.sh
   
5. Then change the permissions to make the script executable in the terminal type: chmod 764 /home/user/search-google.sh
   
6. Then you can create a custom keyboard shortcut to execute the script.
   
7. Go to System Settings > Keyboard > shortcuts > Custom Shorcuts
8. Add a new item and fill the fields Name:search-google and Command:sh/home/user/google-search.sh

9. Then give a keyboard shorcut to the script...click in "Disabled" and press the keys to the shortcut. eg: Ctrl+Alt+G

10. So, if you are in a web page or any document and you select or (text highlighted) a word or phrase... then press the keys for the shortcut eg: Ctrl+Alt+G and Firefox will open searching in google for the text you have highlighted o selected.

On the other hand I think the user55822 tried to say was the Artha
application works like this command.. the difference is by default if you have a text selected with Ctrl+Alt+W Artha will serch for that text but "offline" (As far as I know.)
But you can use both, if you need to search in Artha select the text and press the
shorcut keys Ctrl+Alt+W, but if you want to look in google "online" you can use the script with the shorcut Ctrl+Alt+G.

All the credit for the command is for user55822

Hope will be useful.

Edit: The above instructions on setting the shortcut are for Ubuntu (proper). To set the shortcut for that script (or for any script like that) in Xubuntu, see the answers to this question. To set the shortcut in Lubuntu (which does not have a GUI for that purpose):

Paste in Terminal:

sudo gedit ~/.config/openbox/lubuntu-rc.xml 

There are a lot of groups of lines that set the keybinds and look like so:

<keybind key="C-A-t">
      <action name="Execute">
        <command>x-terminal-emulator</command>
      </action>
    </keybind>

To add a keybind for the google search script, insert this at the end of one of those groups:

<keybind key="C-A-g">
      <action name="Execute">
        <command>sh ~/search-google.sh</command>
      </action>
    </keybind>