To answer your question you must first decide your end goal. Are you growing for the tropical look and don't really care about fruit? If so, leave the pups alone and let the mat grow. Water and fertilize well for best look. Remove pups that grow outside your boundaries based on appearance.

Are you looking to produce the maximum amount of fruit per plant (and live in a climate where you can assure 18-24 or so months of growth until harvest)? If so, you need to reduce competition for water and nutrients. Thin the mat to one mother plant (fruiting or of fruiting size), one about half that age (the daughter), and a pup just emerging from the soil (call this the grand daughter). If your soil is very fertile and you are generous with water and fertilizer you may be able to have a few more active pstems per mat.

I understand thinning the mat to one mother plant (mine is currently fruiting), one half that age, and one just emerging.

My question is about the pups I remove during the thinning...can I form another "mat" with them; plant them in separate pots; or do I have to let them go?

And, how exactly do I separate the pups? I don't want to damage the mother or surrounding plants.
Also my first banana plant.