He turned them on the heavens and immediately made a number of significant discoveries. When he looked at Jupiter, he say four tiny dim lights near Jupiter. But as Jupiter altered its position with respect to the stars, these always stayed near Jupiter. He realised that these must be moons of Jupiter which orbited around that Planet, not stars. Thus not only earth had a moon but Jupiter did as well. Those moons were not orbiting the earth, so things could orbit around something other than the earth. He called these the Medician moons, to honour the 4 brothers of the Medici family, who he was trying to get a job from. The moons of Jupiter were also seen by Simon Marius at about the same time, and his naming of the moons eventually dominated.
Moons of Jupiter as sketched by Galileo
See also http://sbo.colorado.edu/SBO_OLD_SITE/sbo/astroinfo/galileo/galileo.html for an analysis why he saw a variable number of moons on successive nights. and https://en.wikipedia.org/wiki/Galilean_moons for a fuller description of the discovery and of the moons.
The second discovery came when he looked at Venus. As the nights went by, the shape of Venus altered. It looked just like the phases of the moon. When Venus was near the sun, it was either an almost fully illuminated circle, or an almost vanishing crescent. When Venus was far from the sun, it looked more line a half moon. Also the size of Venus changed, from a tiny almost fully illuminated circle to a large crescent, about 5 times larger than the fully illuminated circle. Thus the distance from the earth to Venus must be changing by about a factor of 5. Ptolemy had stuck Venus' orbit so that it lay completely on one side of the sun. In that case one would expect either that Venus would look like the phases of the moon near new moon (ie, a bright crescent which to a something that looked like an almost half moon) or like the phases near the full moon. You would never have a transition between phases near the new and phases near the full moon. His observations meant that Venus could not have its orbit such that it was always further away than the sun, or always nearer. Venus orbit must pass from nearer to further than the sun. Now Ptolemy had no reason for locating Venus (or any of the planets orbits ) at any particular distance from the earth. It was whim how large he made the orbits. His belief in Aristotle's crystal spheres meant he did not want any planet penetrating the sphere of any other planet or the sun, but those crystal spheres were not in any sense and essential part of his model. But since in either Copernicus' or Brahe's models the planets orbited the sun and went from nearer to further away than the sun, this observation of Galileo's strengthened the belief that Copernicus had been right. Venus orbits the sun. Whether the sun also orbits the earth, or the earth orbits the sun, these sightings could not decide between.
Phases of Venus, and drawings of Mars, Jupiter and Saturn from Galileo's notebooks. See https://catalogue.museogalileo.it/indepth/PhasesVenus.html.
Finally, one of the beliefs of the Greeks was that the heavens were a more perfect place than the earth, they had a belief that the heavens should be more perfect than the earth, and not subject to the corruption and change so typical of the earth. Pointing his telescope at the moon, he saw a moon which was not only mottled in appearance but seemed to have projections on its surface, which became more visible when the sun shone obliquely on the features. They seemed like mountains. The face of moon seemed to be as rugged and featured as the earth.
Moon as seen by Galileo through telescope. https://catalogue.museogalileo.it/gallery/GalileoGalileiSidereusNunciusFacsimile.html
All of these observations were strongly suggestive that Copernicus had things right. Furthermore, the Church, via some of the Jesuit astronomers who were in the pay of the church, rapidly confirmed all of these observations. Galileo was eventually put under house arrest, more because he continually insulted the church leaders, then because of the observations or even his views of the heavens.
The telescope rapidly became the primary tool for extending the understanding of the heavens. Huygens, in Holland about 30 years later used the telescope to estimate the distance to Venus, and thus, to the sun. Looking through the stronger telescopes, it was clear that Venus was a round sphere, like the earth as seen in the telescope. Measuring the size of the sphere as seen through the telescope and knowing the magnification of he telescope, he could figure out what the angular size of Venus was (it is about 1/4 of an arc minute-- a bit less than 1/100 of the angular size of the moon.) He then guessed that Venus was about the same size as the earth. and thus could figure out how far away it was, and thus how far away the sun was. This gave a figure about 40 times larger than Hipparchus and Ptolemy and Aristarchus had gotten (about 10000 times the diameter of the earth.) Of course he had no reason for believing that Venus was the same size as the Earth, instead of say the moon, or any other size. it just so happens that Venus is only about 5% smaller than the earth, so his figure was close to the right value.
It is however interesting that such a crucial figure as the distance of the earth to the sun was only found about 350 years ago, when people had been interested in it for over 2000 years.
Projecting an image of the sun, Galileo also saw sunspots on the surface of the sun. Furthermore those sunspots moved, giving support to the idea that sun rotated. Since whether or not the earth rotated was still a hotly contested idea, this also lent support to the idea that the earth rotates, rather than that the whole sky rotates about the earth. (Note that Tycho Brahe still believed strongly that the earth does rotate.)
copyright W Unruh (2018)