Glass was discovered in the first 100 years AD, including the chance discovery of lenses. This occurred with it was observed that glass that is thicker in the middle than at the periphery magnifies objects viewed through it. This was the first biconvex lens.
Microscopes used today are called compound microscopes. The compound microscope has an objective lens that sits very near the specimen, and a long tube and body containing intermediate lenses, and an eyepiece with an ocular lens.
In 1590, two spectacle makers from Holland, Jenssen and Zaccharias invented the simplest compound microscope, consisting of a tube with lenses at either end.
It was Galileo who used the Dutch invention and did extensive experiments. He made significant improvements on the microscope design. He also built telescopes, and published widely.
I believe that Galileo should be given the credit for having the greatest impetus to the use of the microscope to study living things.