The Ring was created by Sauron, and he used his own power to create it. Thus, Sauron wouldn’t change at all himself by wearing the Ring, because the Ring is already part of him. It made him more powerful whilst wearing it, as he could then control the power of the Seven and the Nine (and the Three as well, I believe). Without the Ring, however, his power was significantly weakened.
The invisibility trait is cast upon to mortals who wear the Ring. It takes their form and casts it to the otherworld. Since Sauron is immortal and can already interact with both the physical and “otherworld”, no dice.
It’s interesting to note that the Ring of Power can help you rally armies, but doesn’t make you omnipotent and can’t prevent your own physical destruction. Sauron was defeated three times while wearing the Ring: By Gil-Galad, by Ar-Pharazon (the Numenorians turned his own army against him!), and by Elendil and (a different) Gil-Galad.