Johannes Kepler, an astronomer, suggested in 1611 that using convex lenses within the eyepiece and a convex lens in the objective would help improve the image.  However, Kepler never attempted to make this microscope.  Galileo is credited as the first scientific user of the microscope and coined the name "microscope" in 1624. Christopher Scheiner in 1628 constructed a microscope based on Kepler's suggestions, which is considered the prototype of the modern compound microscope.   

Christian Huygens

Christian Huygens, in 1655, perfected the grinding and polishing of lenses for the eyepiece which improved the ability to see an object without too much color aberration (achromatic lenses).  He also developed what is now termed “Huygens’ principle,” which proposed that light traveled in waves. 

In 1673, Antonie van Leeuwenhoek,  a Dutch storeowner turned microbiologist, perfected a simple microscope with a very short focus  that was preferable to the compound microscopes that were available at that time.  His single lenses were able to magnify up to 300X.  He would construct a simple microscope for each specimen that he observed.  He discovered bacteria, sperm cells, and many other microscopic creatures.  He meticulously recorded his observations in volumes of notes.  Although he was not university educated, he was recognized as a leading scientist of his time.