Galileo Galilei was the first astronomer to use a telescope to study the heavens. Galileo made a number of observations that finally helped convince people that the Sun-centered solar system model (the heliocentric model), as proposed by Copernicus, was correct. These arguments can be divided into two kinds: Those that proved that the Ptolemaic model was incorrect; and those that undermined the broader philosophy of Aristotelianism that included the Ptolemaic model. We'll first consider some philosophically important observations and then the ones that proved Venus, at least, goes around the Sun and not around Earth.
Sun and Moon
One of the ideas that made Aristotelianism popular with the church during the middle ages was that the heavens are perfect. This also meant that they were unchanging, because if they change then either they weren't perfect before or they won't be perfect after the change.
Galileo discovered spots on the Sun and also saw that the surface
of the Moon was rough. People really tried hard to account for these
observations without making the heavens imperfect; one suggestion
was that over the mountains of the Moon there was a layer of clear
crystal so the final surface would be smooth and perfect!
Galileo saw near Jupiter what he first thought to be stars. When he realized that the stars were actually going around Jupiter, it negated a major argument of the Ptolemaic model. Not only did this mean that the Earth could not be the only center of motion, but also it knocked a hole in another argument. The supporters of the Ptolemaic model argued that if the Earth were moving through space, the Moon would be left behind. Galileos observations showed that the moons of Jupiter were not being left behind as Jupiter moved.
Phases of Venus
One observation definitely disproved the Ptolemaic model, although it didn't prove that Copernicus was right (as Tycho Brahe pointed out). This was the observation that Venus has phases, much like our Moon does.
To the naked eye, Venus always appears as a bright dot in the sky. With a telescope, however, it is fairly easy to see the phases of Venus. Just as the Moon has phases, Venus too has phases based on the planets position relative to us and the Sun.
There was no way for the Ptolemaic model (Earth centered solar system)
to account for these phases. They can only occur as Galileo saw them
if Venus is circling the Sun, not the Earth.