Civil Services papers have two broad areas - static and dynamic (mainly current events related).
When you read the newspaper, always keep in mind that - GS papers are of 'General Knowledge', i.e. something which a well read mature person should know. So, my advice is that - while you should read the newspaper thoroughly and try to understand the issue and concepts at hand, never do a deep research as it may be very time consuming and may not be significant from the point of view of the exam.
Secondly, the question may arise - Which newspaper(s) and magazines? How many newspapers should I read? I would suggest that you pick only one newspaper as you have limited number of hours in your day and all the papers cover more or less same stories (especially the significant ones). The Hindu and Indian Express are two good newspapers primarily because they are limited and concise in their content and don't have many distractions and 'extra' news as in case of other tabloid styled newspapers like ToI and HT. My favorite was The Indian Express as content wise it is generally richer than The Hindu (It carries a readable Op-Ed page also apart from a page called 'I.E. Explained' which covers contemporary issues of importance in depth). Also try to read the readers' comment daily. They often give interesting insights and give you an alternative perspective than the column writers. You may also start thinking critically and may send a comment once in a while. It can be a sort of answer writing practice for you (and since in this case you will be read by millions of other readers, you will automatically more careful in writing your thoughts :) ).
Thirdly, the question may arise - What to read in the newspaper(s) and magazines? To get an idea, read last years question papers again and again to enter into the psyche of the paper setters. You will slowly realize that often conceptual things are asked and you need to learn only basic facts. Further, you will also realize that development of critical thinking is an integral part of answer writing. So, whenever you read, give a critical thought to the item you read.
Fourthly, I would also suggest that you also try to search the issue on internet or Wikipedia. Often, there are Wiki pages on all the major issues (you will find wiki pages on every possible topic - say for example, try searching Indo-China relations) which give background, causes, consequences and so on of a particular event.
Finally, never keep cuttings of newspapers or magazines (and similarly, save the raw articles from internet) as over the time they become so bulky that you will never be able to even skim through them during exam time. make your own notes topic wise and try to find every dimension of it (background, players involved, significance for India/society/World, future possibilities and so on). By doing so, you will be kinda writing a comprehensive answer on a particular topic and you will save time on separate answer writing practice.
Here are some of my notes on current events and other topics -
These notes will give you an idea that you should organised your notes so that you don't have to save the same article many a times. You may simply make new addition to the topics already covered in such notes.
Leave a comment if you notice some discrepancy in these notes or have some suggestion or query.
Best of luck!