I think there are a couple of 'transitions' from school to university/job hunting which are crucial to get right and which can often make the difference between success or failure in getting graduate work.
The first is the de-emphasising of marks. At school, the focus is on getting the grades that you need to do what you want to do at university - mostly, if you get those grades, you get your place, so it's all about marks and getting the highest possible mark. The graduate job market is quite different - most graduate jobs require a 2(I) degree or higher, but most graduates get that. I find some students remarkably resistant to the idea that if they are getting solid 2(1) marks, their priority needs to be on extra curriculars to develop skills and experience, rather than pushing up to a first - there are very few circumstances where having a first rather than a 2(1) will make a material difference to your chances of employment. Internships help, but so do other sorts of involvement at university - volunteering, clubs and societies, and generally taking time to be involved and understand about the world you're hoping to be part of. Most universities will have huge amounts of advice and practice available, but making it compulsory just doesn't work - students have to want to make time for it and that's about understanding what is required and that employers want to employ individuals, not degree transcripts.
The other is about having a realistic sense of what you want to do and how to get there. This means moving beyond school level "I want to be an xxx" and understanding what that really means, how many different ways there might be of being an xxx, and what other alternatives to being an xxx might exist that might actually suit better, and planning a route to becoming an xxx which, in many cases, may not be easy. It's about moving beyond thinking of career as something that happens in the future, and realising that university is not the last step of education but the first step of your adult, professional life. Sometimes it's about understanding your first job as your first job, and not the thing you are going to be stuck in for ever - some students seem to get paralysed because they think they have to decide what they are going to have to do for the rest of their lives, when really all they have to do is decide what road they want to go on, and how they are going to support themselves. Very few of us achieve our ambitions at the age of 22, and most of us take lots of twists and turns in life, and sometimes I don't think young people quite see that. The risk is either that they get paralysed into not making applications at all (and if you don't apply, you won't get a job) or they are too selective and want to get to the perfect job straight away, rather than thinking strategically about ways and means.
So in my experience, most students get graduate jobs, but some get them quicker than others and, generally, I can predict who is and isn't going to by seeing how they are approaching their university life.