Studying & Doing Philosophy: How is Philosophy Studied? How is Philosophy Done?

The study of philosophy is usually approached in one of two different ways: the systematic or topical method and the historical or biographical method. Both have their strengths and weaknesses and it is often beneficial to avoid focusing on one to the exclusions of the other, at least whenever possible. For irreligious atheists, though, the focus should probably be more on the topical than on the biographical method because that will provide clear overviews of relevant issues.
The systematic or topical method is based upon addressing philosophy one question at a time. This means taking on an issue of debate and discussing the ways in which philosophers have offered their views and the various approaches they have utilized. In books which use this method you find sections about God, Morality, Knowledge, Government, etc.

Because atheists tend to find themselves engaged in specific debates about the nature of the mind, the existence of gods, the role of religion in government, etc., this topical method will probably prove most useful most of the time. It probably shouldn't be used exclusively, though, because removing philosophers’ answers from their historical and cultural context causes something to get lost. These writings were not, after all, created in a cultural and intellectual vacuum, or solely in the context of other documents on the same topic.

Sometimes, a philosopher’s ideas are best understood when read along with his or her writings on other issues — and that is where the historical or biographical method proves its strengths. This method explains the history of philosophy in a chronological manner, taking each major philosopher, school or period of philosophy in turn and discussing the questions addressed, answers offered, major influences, successes, failures, etc. In books using this method you find presentations of Ancient, Medieval and Modern philosophy, on British Empiricism and American Pragmatism, and so forth. Although this method can seem dry at times, reviewing the sequence of philosophical thought shows how ideas have developed.

Doing Philosophy

One important aspect of the study of philosophy is that it also involves doing philosophy. You don’t need to know how to paint in order to be an art historian, and you don’t need to be a politician in order to study political science, but you do need to know how to do philosophy in order to properly study philosophy. You need to know how to analyze arguments, how to ask good questions, and how to construct your own sound and valid arguments on some philosophical topic. This is especially important for irreligious atheists who want to be able to critique religion or religious beliefs.
Simply memorizing facts and dates from a book isn’t good enough. Simply pointing out things like violence committed in the name of religion isn't good enough. Philosophy depends not so much on regurgitating facts but on understanding — an understanding of ideas, concepts, relationships, and the reasoning process itself. This, in turn, only comes about through an active engagement in the philosophical study, and can only be demonstrated through the sound use of reason and language.
This engagement, of course, starts with understanding the terms and concepts involved. You cannot answer the question “What is the meaning of life?” if you don’t understand what is meant by “meaning.” You cannot answer the question "Does God exist?" if you don't understand what is meant by "God." This requires a precision of language not normally expected in ordinary conversations (and which may at times seem annoying and pedantic), but it is crucial because ordinary language is so rife with ambiguities and inconsistencies. This is why the field of logic has developed a symbolic language for representing the various terms of arguments.

A further step involves investigating the various ways in which the question can be answered. Some potential answers might seem absurd and some very reasonable, but it is important to try and determine what the various positions may be. Without some assurance that you have at least brought up all of the possibilities, you’ll never feel confident that what you have settled on is the most reasonable conclusion. If you're going to look at "Does God exist?" for example, you need to understand how it might be answered in different ways depending on what one means by "God" and "exist."

After that, it is necessary to weigh the arguments for and against the different positions — this is where much philosophical discussion takes place, in supporting and critiquing different arguments. Whatever you finally decide upon will probably not be “right” in any final sense, but by assessing the strengths and weaknesses of the different arguments, you will at least know just how sound your position is and where you need to do further work. Too often, and especially when it comes to debates over religion and theism, people do imagine that they have arrived at final answers with little work done to seriously weigh the various arguments involved.

This is an idealized description of doing philosophy, of course, and it is rare that any one person goes through all of the steps independently and fully. Much of the time, we have to rely upon the work done by colleagues and predecessors; but the more careful and systematic a person is, the closer their work will reflect the above. This means that an irreligious atheist can't be expected to investigate every religious or theistic claim to its utmost, but if they are going to debate any particular claims they should spend at least some time on as many of the steps as possible. Many of the resources on this site are designed to help you go through those steps: defining terms, examining various arguments, weighing those arguments, and reaching some reasonable conclusion based upon the evidence at hand.