How exactly did Galileo prove that Earth goes around the Sun? Could you explain what observations and conclusions he made ?
Galileo used his observations of the phases of Venus to provide evidence that the Earth orbits the Sun.

At the time, the prevailing model of the Solar System was the geocentric model, in which the Earth was believed to be the center of the Universe and all other celestial bodies revolved around it. However, Galileo's observations of Venus showed that it went through phases, just like the Moon. This could only happen if Venus was orbiting the Sun, not the Earth.

In addition to his observations of Venus, Galileo also made other observations that were inconsistent with the geocentric model. For example, he observed four large moons orbiting Jupiter, which could not be explained under the prevailing model.

Galileo's observations were controversial and earned him the ire of the Catholic Church, which adhered to the geocentric model. However, his work helped to pave the way for the modern understanding of the Solar System as a heliocentric system, in which the Sun is at the center and all of the planets orbit around it.

It should be noted that while Galileo's observations were important in establishing the heliocentric model, he was not the first person to propose it. The Polish astronomer Nicolaus Copernicus had proposed a heliocentric model a century earlier, but his ideas were not widely accepted until Galileo's observations provided further evidence to support them.
Thanks, I'm really feeling educated now.
You are welcome. Let me know if there is anything else that I can help you with.