How might I become proficient in historical thinking and historical argument?

If you master historical thinking, then you will be more proficient in making a historical argument.  These things will put you on a path towards becoming a true historian, which means that you possess critical thinking skills and are good at finding cause/effect relationships.  This will help you in any field.  
If one wants to master historical thinking, one should be careful as to what sources they are using.  One should focus on primary sources (materials created by someone from that time period) and look for bias.  Is this person trying to make themselves look good for posterity?  Is this person trying to incriminate others?  Also, do not judge historical people by present-day values.  Attitudes that we now take for granted were not considered normal back then.  One hundred years from now, I am sure that values will change once again and people reading about our time period will laugh and occasionally cringe.  Also, look for the reasons why historical people do things.  Look at what they are trying to accomplish through their words and actions.  
For your historical argument, I think you are having to write a paper.  My advice is to craft a a thesis statement which you can defend, and then find four or five concrete details which back your thesis.  If you do this, you will have identified what you are trying to prove and then displayed evidence, proving your point and demonstrating your mastery of not only the subject but historical argument in general.