The Digital Spy site is working fine on my computer, using IE8.
Compatibility View shows the site as it would have appeared using an older version of IE.
To use it, just click the button and see if the site appears correctly. There should be no need to use Firefox if you don't want to.
I agree with rdave13 that it could be your IE8 shortcut that is causing the problem. It could also be that you IE 'cache' needs clearing. To do this, click on 'Tools' on the IE menu (button with a cog on it). Select 'Internet Options'. On the 'General' tab, click on the 'Delete...' button under 'Browsing History'
To answer your original question - yes you can use both IE8 and FF (and any other browser if you wanted). When you want to browse to Digital Spy just open FF and use that instead of IE8. You can then return to IE8 whenever you want.
You could keep both IE8 and FF open at the same time - just click on the Taskbar at the bottom of the screen to switch between them. IE8 can remain your default browser throughout.