Interacting with the audience in the middle of a dialogue with another puppet often surprises the audience members and makes them laugh. If someone is talking, stop the performance, have the puppet turn directly toward the audience member, wait until they stop talking and have the puppet confront the noisy audience member. A simple, "Excuse me, we are trying to put on a show here" will usually make the audience laugh. It is unexpected and out of the ordinary and when done with humor, it will be taken in the light manner intended.
In the middle of a puppet show, while two other puppets are arguing, have a third puppet enter the scene and begin singing a silly song. The other two puppets should stop what they are doing and look first at the singing puppet and then at the audience, then back to the singing puppet. Think up funny ways for the two non-singing puppets to get the singing puppet to quiet down. As an additional gag, the singing puppet can come out at random points throughout the show. Just don't overuse the singing puppet, as it can become old very quickly.
Copy an old Abbot and Costello script and adapt it for use with puppets. The scenery doesn't need to be very complicated; you can simply have two puppets talking to each other with a nature scene as the background. If you use the "Who's on First" baseball routine, you could create a baseball stadium as a backdrop. To help create emphasis place a strong stress on the names of the people in the routine. In "Who's in First" there is some confusion about the name of the guy standing on first base. Since his name is Who, it creates a humorous confusion between the two characters.
Create a parody or satire of a popular current event. Maybe have two puppets, a Democrat and a Republican, run for office and show the different ways the two parties go about asserting opinions and policies. For added humor, the Democrat can be a donkey and make donkey noises and the Republican can be an elephant and wave his trunk around while talking. These are good choices for puppets since it will be easier for the audience to identify the party by the political party mascot.