How can one improve one's analytical skills?
All skills can be improved both by training and practice. Taking courses which focus on analytic skills are a way to train them. Perhaps the courses most focused on pure analysis are logic courses, including both formal and informal logic, because the entire focus of logic is using reasoning skills. You should also think about taking courses in rhetoric and writing, because as well as teaching skills in argument and analysis, these courses teach you how to express yourself clearly so that that people can understand your ideas and see that you have done a good job of analysis.
For practice, as you read, try to outline the main arguments of the works you are reading to make an analytical skeleton.