What style of music is in Pirates of Penzance?

I'd agree with the excellent answers from the folks above, but I'd disagree that there not recitatives. Pirates does have recitatives here and there, however short. They usually are one or two note affairs that serve as brief passages that take us into larger numbers, but they are there.

"Hold, monsters! Ere your pirate caravanserai proceed against out will to wed us all!"

It's an operetta- it's like an opera, except the singing is closer to musical-theatre style singing (no heavy opera vibratos), and (as the poster above mentioned) there are no recitatives (sung sections that advance the plot instead of providing a song melody). Thus, it is an easy way to familiarize yourself with classical music without getting too deep in it (since it has spoken dialog, you should have no trouble following the story.)

Operetta. Similar to opera but there are few to no recitatives (where dialogue is sung as opposed to spoken).

I'd suggest renting the film with Kevin Klein and Linda Ronstadt.