Yes the kind of wood does matter. I recommend lime wood. But it depends what carving woods are available in the States. Pick up a magazine on woodcarving or whittling off a newstand and see what's available.
Use power tools by all means - whatever you find easier. You'll loose fewer fingers than with chisels. Also you don't have to carve. Try modelling. My own modelled and hollow cast heads have survived forty years of battering. Bryan Clarke, one of the leading Punch figure wood carvers, often used papie mache for his own figures. These can be very strong if made properly.
But if wood is your thing, then by all means have a go. A nice wooden set is very satisfying.
And remember, historically, Punch and Judy profs carved their own figures which were rarely works of art but often had a primitive vigour which came from the carver's lack of sophistication.
Just to inspire you, look at these great figures from a great American puppeteer from around a hundred years ago - these are Punch & Judy used by the Lano family.
Carved on the road, certainly without power tools and probably no chisels or vise and most caravans aren't equipped with a workbench.
