Going into politics?
Step #1: sell your soul.
Ok, in all seriousness, you just want to educate yourself and be able to hold your own in discussions. First thing I’d do is read a bit about government in general; get a book from the library, or a textbook if you still have one, and review how government is structured and how it’s supposed to work. If you really want to get down and dirty, read the Constitution, the Federalist Papers, and other works that explain the origins of our government and its early history. But a general text/survey will do, just to get the basics.
The second step is to read, read, read newspapers, news sites, blogs, etc. that cover news or analyze news. Get balance, though; read a few U.S. newspapers/sites, and then check out foreign news sites for a different perspective (the Guardian, BBC, Al-Jazeera are just a few you could check). Additionally, depending on your political bent, read some newspapers/sites that are slanted towards opposing beliefs (as @Jaxk pointed out). This will help you understand the central arguments people are making, regardless of whether you agree with those arguments (or even find them valid).
As @filmfann states, knowing some history and economics will help a lot as well.
Observe other discussions/arguments. Over time, the knowledge you’ve accumulated will help you pick apart the points people make, and once you’re able to deconstruct someone’s arguments, you then are capable of making your own. At that point, jump in. You’ll then be on the offensive and defensive, and learn as you go along how to craft your arguments. You’ll also know when to be diplomatic, when to shut up, and when to let go.
Politics inflames people, but since there is politics in everything, the more you learn about how politics works, the more you’re able to engage with others around you and understand how our society works (or doesn’t work, for that matter!).