Firefox does that by default. You probably have an add-on installed which changed the focus of the cursor from the URL bar to the Search bar.

http://support.mozilla.com/en-US/kb/troubleshooting+extensions+and+themes

By default Firefox should go to the URL bar automatically. Have you changed any settings? A reset may fix this by going to "Help">"Troubleshooting Information" and clicking "Reset Firefox". If this doesn't work, don't hesitate to to keep asking questions.

What is the URL of the about:newtab page (pref: browser.newtab.url) ?

Firefox goes to the search bar on the about:home page and may also do this for some other pages.

Had the same problem with FF 17 update. It had to do with Tab Utility or Tab Mix Plus. I worked around it by installing an add-on called InstantFox for searching from the URL.

Worked for me.

If you are using the NewTabURL Add-on click on Tools >> Addons >> Scroll to the NewTabURL > Click Options.

Near the bottom is a check box (checked by default, I think) Select location bar after loading new tab.

Make sure it is checked.

In my case, it must have been the add-ons. After uninstalling all of them, cursor was placed in the adress bar when opening a new tab.

I've set my new tab page to be about:home. On that particular page, Firefox defaults to the search box instead of the URL bar. Is there any way to change that setting?

Pressing the F6 key should bring the focus to the location bar in such a case.

Sure, and I use Ctrl+l (that's a small L) currently. But I was asking about the cursor defaulting to the URL (awesome) bar.

The search bar on the about:home page gets focus if that URL is used for the new tab page.
You will have to use an internet page as the new tab page if you want the focus on the location bar.