There's some confusion in your question between heliocentrism and the idea the world is round.

Heliocentrism is the idea that the Earth goes around the Sun, not that the Earth is round.

People in Western Europe knew the world was round. But they also thought that the Earth was the centre of the universe. There was some Bible verses that could be seen as supporting this, but the main issue was that it seemed to fit the evidence, to the astronomers at the time. The current models of Physics used couldn't explain the movement of bodies like the Earth, so it was assumed the stars and planets must be fundamentally different and made of different materials (aether). There was also the issue that the stars don't appear to have what is called Stellar Parralax, ie move in relation to the stars behind them (unless you're using a really good telescope). Their ideas went back to Greek astronomers, particularly Ptolemy.

Copernicus was the first European in the Middle Ages to suggest a Heliocentric model. However, he was unable to offer any conclusive proof, particularly on the subject of Stellar Parallax, so his book was considered interesting, but didn't have a massive impact. It's also noteable that he dedicated his book, published in 1543, to the Pope, saying it would be useful for the reforming of the Julian calendar, and included a letter from a theologian saying the theory had no theological ramificaitons, and a letter from a Cardinal urging the work to be published.

Galileo was a very respected scientist, but he could be very rude to his opponents. He had a huge argument with the Jesuit scientist Orazio Grassi about comets, publishing his withering views about Grassi's scientific credentials. This turned the Jesuits against him, but the man who was later to be Pope, Urban VIII, was a big fan.

Galileo, having a more advanced telescope, discovered 4 moons of Jupiter. This was the first time anyone had proven the existence of things that definitely didn't orbit the Earth. So, this made him interested in Copernicus' theory and he started writing in support of the theory.

In 1613 the Grand Duchess of Florence got into a (civilised) argument with one of his supporters, about whether the Bible supported Geocentrism. This caused Galileo to write a letter giving his ideas about the Bible and science, and using theology to argue that the Bible did not contradict Heliocentrism.

This caused the Inquisition to look into the matter. Although the opinion wasn't unanimous, they eventually declared, having consulted with other scientists of the time, that Heliocentrism was unsupported and should not be taught, and Galileo's book was banned.

For about 10 years after this, Galileo stayed out of the debate, until his friend became Pope Urban VIII. Urban encouraged him to publish a new book giving fair and balanced evidence for and against Heliocentrism, with the Inquisition's approval. He also asked him to include Urban's own opinion.

Unfortunately, the character in the book who gave Urban's geocentric opinion came off as rather stupid, and was named "Simplicitus". It seemed therefore, that rather than giving a fair and balanced view of both sides, Galileo had written a book insulting the Pope and other geocentrists, saying they were idiots.

So, the Inquisition arrested Galileo again, investigated, and once again condemned him. This time, they sentenced him to life under house arrest. He spent his time writing books about kinematics and materials science.

And later on, we got better telescopes, discovered stellar paralax, and it turned out Copernicus and Galileo were right after all.

tl;dr, Heliocentrism was originally seen as an unthreatening but unsupported idea, and Copernicus took care to explain that it had nothing to do with theology. But Galileo got into trouble by writing about theology to support his theory (that was still unproven), then compounded it by (probably accidentally) insulting the Pope. Remember that this wasn't all that long after the Reformation, so the Church tended to react very badly to laymen writing about theology.