Persuasion is how the Liberal Crime Squad grows and prospers. The skill is based on & limited by the Liberal's Charisma.
Applications of Skill
Persuasion is the default social skill. It governs talking about the issues on-site, evening political meetings (dates involve Seduction), taking over media stations, defending oneself in court (along with Law), and tampering with Juries (again, combined with Law). In combination with Street Sense and Business, Persuasion affects distributing fine Liberal baked goods, as well.
Training this Skill
Just about any use of Persuasion trains it; the safest way to build points is to find Liberal members of society and talk about the issues. Successful evening meetings will boost the skill. Highschool Dropouts and Hippies are generally the easiest folks for low-Persuasion Liberals to impress.
Persuasive Liberals can train their colleagues via teaching Political Activism, and the Public Policy coursework at the University can help with the first few points.
Bonuses to Skill
In practice, the only way to improve one's Persuasion is to do more of it and/or gain Charisma.