The difficult thing about learning how to make art with social commentary is that the subject is so broad. Most representational art, and even some abstract art, contains social commentary in some form. Whether you are using your art to protest a war, glorify a certain image of beauty or lampoon a certain common attitude you object to, you are making some form of social commentary. The difference is when your intention is to inspire social change you have to make sure that the commentary is at the forefront of the art. Make it obvious enough for people to get it without insulting your viewer's intelligence.
Decide on what you want to say. Some popular topics in political art today are environmental destruction and warmongering, but art can also speak about gender roles, public space and education.
Think about how you want to make your artistic point. Picasso, in his mural Guernica, used his own inventive painting techniques to depict the Nazi bombing of Guernica. The message was conveyed through the art, which was as important as the message. The Guerrilla Girls, by contrast, use slogans, advertisements and performance art to protest sexism in the art world. They are much more concerned with the content of the message than the artistry of the message. The graffiti artist Banksy takes his social commentary one step further by spray-painting it on public spaces, generally without permission. He comments on the use and control of public space in society (among other social and political issues) by physically taking that space. Do you want to make a statement in your art or use your art as a means to spread your statement? The distinction is subtle, but important.
Decide on a medium. Ask yourself if you want to draw, paint, sculpt, combine multiple media, or use something less traditional. Film, art installation or performance are options you might want to consider. Find something that speaks to you.
Plan it out. Draw some sketches or, if you are staging a performance, write out a script. Take as much time as it takes to hammer out your ideas.
Steal what you need. Don't rip off someone else's idea wholesale, but "borrow" any elements that inspire you and make them your own. You can find inspiration in fine art, advertising, urban layout and popular television shows. Taking recognizable images and reworking them establishes a direct relationship between your art and the world around you.
Make your art. The process will vary widely depending on what you are doing. For a performance art group such as the Surveillance Camera Players, for example, making art means writing out a script, making a few simple posters and acting it out in front of surveillance cameras and whatever bystanders stop by to watch. For a fine artist such as an oil painter, however, the process will be much more protracted and detail-oriented.
Keep the message pretty simple. Complex messages work well in writing, but art usually works best with simple messages. Be seen. No matter how good social commentary is, it needs to be seen to have any effect. Study the works of other artists. No matter what kind of art you are making, you will always benefit from exposure to new ideas and new techniques.
If you do decide to do graffiti art without permission, don't stick around to meet your admirers. Remember, the cops won't think of it as art.