I work for a documentary and reality TV company, have a few fiction projects in development. I'm no big shot by any stretch, but I sorta know how things work. TV and film companies don't care about degrees, they only care about experience. However university years were some of the best years of my life, so it's not to be discounted. I did a computers degree and a business masters and those teach you a lot of skills that employers are impressed by. A lot of film courses are watching films about soviet silent cinema which is of no use to employers. If you've got a real or useful degree you're a much more attractive prospect and you've got a safety net if you decide media ain't for you (which I've seen happen a lot)
Now this doesn't mean you don't make films. Cameras are cheap and easily blag-able. If you want to make films, you'll do it in your spare time. If you're not willing to do it now when you've got a lot of free time, then you'll probably not cope well with the demands of actually working full time for a media firm. You're competing in one of the most desirable industries in the world, so there'll always be people willing to work their asses off. That said if you find filmmaking fun then it's not so much of a chore. Make a shitload of films, shorts, sketches, music videos, whatever. You'll learn lighting, camera, editing, production, scheduling, budget, blagging favours and all the practical shit that people actually need. It'll kill your social life, leave you sleep deprived and possibly cost you some mates, but you'll meet a load of interesting people and it's rewarding.
Oh and learn writing. Everything from directing live action, to character design in animation to editing is helped by knowing writing. As with all filmmaking stuff there's tons of information online, particularly the following essentials: Jane Espenson's blog, the creative screenwriting magazine podcast and wordplayer. Read Field and McKee and you won't agree with everything but at the very least you'll know what everyone else is doing.
Erm, cheeky plug for something I'm working on for a competition that starts on February 23rd.
http://www.youtube.com/watch?v=PM6YbzlGF_w - Trailer.
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Cheers. Sorry for the rambly post, hope there's some useful advice in there.
Eoin