How does one film a puppet show?

I am taking on a project of filming puppets. This is a short 3-minute scene, so I will not have an elaborate set or many puppets. I would like to have a couple of backdrops to represent the single scene, including windows and doors which will contain other puppets.

There will be speaking and singing.

I do not have access to a professional studio, but I have access to some decent video cameras.

Is it best to film everything at once? Would it be a good idea to film the soundtrack and speech first and then record the video? I'll have access to some film-editing software, so I can overlay the audio and possibly slow down the video to match the audio. I envision filming the puppets with the audio playing in the background, so the puppeteers can synchronize their movements to the audio.

This is all quite new to me, so I appreciate any advice.