How do we know how far away the stars are? (serious question)
(Another serious question.)
http://curious.astro.cornell.edu/about-us/41-our-solar-syste...
Knowing those distances, the distance to the sun becomes a trigonometry problem.
Now, we use certain types of stars like pulsars that have known brightness for the particular frequencies. Once we find a new pulsar that matches, we can compare how bright it is to others of the same type. If it is brighter, it is closer. If it is dimmer, it is further away. I'm way way oversimplifying it, but hopefully it gets the concept across.
https://www.amazon.com/Measuring-Cosmos-Scientists-Discovere...
1. It actually required more epicycles than the Ptolemaic model.
2. It didn't even have the other planets orbit the Sun! Rather, they orbited the center of the Earth's orbit (???).
Kepler's laws fixed all of this, but Galileo ignored all of Kepler's correspondence, except for one time when he found it profitable to use him: He had made claims about the Jovian moons based on observations through his telescope, but he refused to let other astronomers borrow that instrument so they could reproduce his observations. Kepler wrote to Galileo offering to defend him, and Galileo accepted. Then later, after Kepler had already publicly defended him and himself requested access to the telescope to verify the observations, he was ignored, as per all other times Kepler wrote him before and after.
However, the barycenter of any sun/planet 2-body system besides Jupiter lies within the sun's surface, so I'm quite comfortable with the claim that at the very least most planets do orbit the sun...
To an extent those in the upper echelons of human society have always known the magical stories were bogus. But when the flock is dumber than Humperdoo himself, Humperdoo will do.
Heliocentrism made a specific prediction of parallax --- namely that the stars would move back and forth over the course of a year. But this wasn't observed which is one of the main reasons that many astronomers remained skeptical of the idea. Tycho Brahe proposed an alternative model that adopted many of the advantages of heliocentrism but didn't have the parallax problem. In his theory all the planets revolved around the Sun, but the Sun revolved around the Earth.
Of course it turned out that the stars were much, much further away than anyone expected, so the parallax effect was too small to be seen until the 19th century.
But your broader point is correct, Galileo's troubles were brought about more due to his political indiscretions than his astronomy per se.