Can I open a new tab but stay on my current tab instead of automatically jumping to the new one?

Click the Firefox button, go to Options | Options | Tabs.

Remove the checkmark from the box which says: "When I open a link in a new tab, switch to it immediately".

See also: Tab options

Sorry if I was unclear but by "I haven't selected the option of automatically switching to new tabs either" I meant I had already unchecked that box and it still jumps automatically to a new tab when I open it, it does that even when I've unchecked it in safe mode. I just want to remain in my current tab and be able to open new tabs in the background. Thanks.

Try running Firefox in Use Troubleshoot Mode in Firefox. If it functions properly in that configuration, then one of your add-ons is the culprit.

"I've tried safe mode and it wasn't any of my add-ons that are the problem and I haven't selected the option of automatically switching to new tabs either."

I've just tested it myself and I see that what you say is correct. I think it's a bug.

Actually, the setting does work correctly. What you need to do here after changing the setting is to clear the cache. I'd shutdown my machine overnight last night and had forgotten to re-enable the option to switch to a new tab immediately. Consequently, while reading BBCNews.com this morning, a link didn't switch immediately when I clicked it.

To clear the cache, click the Firefox button, go to Options | Options | Advanced and in the Network tab, Offline Storage menu, click the Clear Now button.