Hi there, welcome to the forum.
First of all, going to university doesn't automatically mean you're ensured or even entitled to a job in the design industry. I know many people who have completed a graphic design degree, and have been unable to find a design related job, and instead settled for jobs in Asda. The design industry is a pretty tricky one to get into, successfully anyway. There is so much competition out there, some good but most bad, that you really do have to work hard, whether you have a degree in it or not.
Whilst at university, I did a work placement with a guy who set up his own design studio in Bristol. He didn't even go to college to study design, let alone university. But he enjoyed drawing, decided he wanted to become a designer, and had the drive to set up his own agency. And he did.
It's all about experience, communication and showing what you can do and why people should hire you. Yes, some people may well have a degree in the subject, but if their work is shit, lacks imagination, or creativity, they're not going anywhere.
So, building your portfolio is the first thing to do. To begin with, just make up some briefs and do them yourself. That's what I did. I designed logos to briefs I came up with sat on the toilet. They may not have been real, but they showed I could design a logo based on a set of instructions / guidelines. I then had some work I could show people.
Alternatively, if you find that uninspiring, try finding unpaid work. Unless you're super lucky, no-one will pay you hundreds of pounds right away. If you know people who run a business, just ask them if they'd like a logo done, or some posters, flyers, whatever, just offer your services.
If you don't know anyone, walk down your high street, pop into some shops and offer your services. Chances are, most will say no, but even if 2 or 3 say yes, it's a start.
Eventually you'll find people willing to pay small amounts for work. Some payment is better than none, to begin with anyway.
So firstly, build up a decent portfolio. You don't need to jam it full of logos. Pick a selection of your best bits, maybe 6 or 7 to show what you can do. Try and use a selection of work, logos, print based stuff, illustrations, web design, whatever.
Once you have a nice portfolio, you need to get out and talk to people. Ring businesses, email them, walk in their doors and speak to them. Tell them you're a designer and want to work with them.
Why not try finding a design based work placement. Email or ring some local design agencies, and ask if they'd take a junior designer on for a week. You will get experience and knowledge and learn a bloody huge amount. If they like you and what you do, they may even offer you a job. Who knows.
But, experience and communication. 2 things every young designer needs to get anywhere in the design industry.