If you want a first class degree you need to go above and beyond what you are being taught / told to read. The problem with many University courses is that they are often a little out of date, certainly many books in the library will be.
Try reading specialist publications relevant to your degree, usually available in the university library, if not, subscribe.
Speak to your university lecturers, try and find out what direction a particular field is moving in, what is on the horizon?
Using the Internet is invaluable in this process enabling you to:
- Find contemporary academic papers relevant to your degree subject
- Join discussion forums relevant to your degree subject
- Discover the latest / most recently published developments in your field
The Internet, can, if used properly, put you on the cutting edge in terms of your knowledge of current research and thinking.
It can allow you to join debates regarding current and future developments relevant to your degree subject.
Discussing your degree subject and express your own opinion’s on it really gets you thinking.
It is this kind of advanced up to date knowledge and thoughtful debate that can give you the edge required for a first class degree.
Be selective, there is a lot of rubbish published on the Internet (hopefully not this site 
), and over reliance on electronic sources is frowned upon by Universities.
You still need to find a good quantity of recognized texts to support your arguments within degree assignments.
Get as deeply involved with your degree subject as you can, this will not only help you excel in degree assignments, but can really help generate ideas for your dissertation.
[...] Cottoning on I would say comes down to widening your research and engaging with lecturers, finding out what is required and benchmarking against other peoples degree work. I am not sure what is meant by “reading in the right way”, my interpretation would be that you must learn to pick out what is relevant to your degree, but also maintain balance and not take information out of context [...]