I am familiar with the random cards and the carnival, but I don’t know all that much about how the new Renown and Connected mechanics. I’m thinking about this as PoSI is around the corner and I need 5 Society Renown for the Duchess Audience.
There is an excellent guide http://community.failbettergames.com/topic24895-the-ultimate-guide-to-favours-and-renown.aspx
Up to 5 Renown in anything is easy: go to the Carnival’s Big Top or Sideshows. The only thing limiting you is the need to spend favours (because you can’t go over 7) and some quite easy checks.
Past 5 gets hard, but 5 doesn’t require cards or special renown items at all.