lulz You're not stupid. All of us frequently fall prey to The Great Satan that is modern computing!
Are you using an add-on that gives you the Google search box you mentioned? FF, of course, has a built in search box on the top right, but I wasn't aware of an option to make it open searches in a new tab. See, it may be I who is learning here. xD
If I go to ... Tools >> Options >> Tabs ... then I see five check box options:
1.) Open new windows in a new tab instead.
2.) Warn me when closing multiple tabs.
3.) Warn me when opening multiple tabs might slow down Firefox.
4.) Always show the tab bar.
5.) When I open a link in a new tab, switch to it immediately.
None of these five options seem to be related to the built in search bar. Where is the little check box to open searches in a new tab you mentioned?