[This page is a rough idea for now. If you leave suggestions in the comments, I may work them in.]
Here are some my favorite resources on improving your communication skills:
Cognitive Politics: a Communication Workbook for Progressives gives a short introduction to many approaches to communication. Dig deeper or take the time to practice:
Broad Communication & Marketing Skills
I've found reading almost any business or therapeutic book on communication while thinking about politics can be very powerful.
* Difficult Communications
Communication and Conflict
Taking the War Out of Our Words.
Search out Nonviolent Communications (NVC) workshops near you. There are many communities aiming to make a different way of communicating part of life — gentler but not afraid to deal with conflict either.
Livingroom Conversations is a great way to jump in and practice, gathering people with different ideological positions who all still want to connect.
Framing Your Perspective
George Lakoff is very famous among progressives for his advice on word choice.
[Chapter 1 of Cognitive Politics]
Understanding Other Perspectives
Jonathan Haidt's The Righteous Mind
Strangers in Their Own Land