First, understand and learn the various question types. To broadly list the types, they are - Strengthen/Weaken the argument, Evaluate, Inference, Complete the argument, Boldface, find the Assumption, find the flaw, paradox etc. It is very important to understand what the question stem is asking before understanding the argument. (The Manhattan GMAT guide for critical reasoning helped me a lot to improve on this. )
Initially, try to make a quick summary of the argument on paper. Write down only the most important aspects of the argument, using short-hand, signs and variables. Try to identify the premise (facts, supporting statements, information) and the conclusion of the argument. Most of the time the question stem expects you to work on the conclusion of the argument. Track down and attack the conclusion. Start with some pre-thinking before looking at the options, for e.g. What will strengthen the conclusion ? What acts as a bridge between the facts and the conclusion (For assumption questions) ? What could be the reason for such unexpected finding ? During the CR question, remember, that only the information provided by the argument is to be used to work on the argument.
Once you get the general idea about the argument and the question stem, only then start looking at the options (Do not hurry in this process, else you may end up going back and forth between the argument and options). Try to eliminate 2-3 incorrect options. Such options are out-of-scope, irrelevant, going against the question stem or plainly absurd. When left with last 2 options, refer to the conclusion, question stem (from your summary) and take a wise decision. This process can be completed in less than 2 minutes but only with practice. (I got 90% accuracy with this process)
Initially try to master these CR question types, in this order : Strengthen, Weaken, Paradox, Inference, Assumption. These question type will more or less make up for 80% of your CR questions on the test (as per my observation). Do not fret over complicated hard CR questions in the beginning. Start practicing with easy and medium level questions first.
Some quick strategy points :
Strengthen, Weaken and Paradox question options may contain new information, do not eliminate unless you are quite sure.
Assumption options bridge the gap between facts and conclusion of the argument. Negation test : If you negate the correct answer , it will weaken the conclusion completely. This works only on assumption questions
Evaluate : Try to find the answer which has maximum impact on the conclusion i.e. it can both strengthen/weaken the conclusion.
This is just some basic idea to start with CR. I used to dread CR questions but with enough practice, it became my strength in Verbal (Along with SC).