One was over the existence of the congress. Some believed that only a senate should exist while some believed that only the house should exist. They made the government bicameral then to equally protect all states, regardless of their size or their population.
Another was then over slavery. Since slaves weren't considered citizens or even people, it was debated how they should be accounted for in the census for representation. This was solved with the 3/5ths compromise which meant that the southern states got political power in the house of representatives even more.
Another issue was about protecting civil liberties. Those who opposed the constitution claimed that civil and state rights and liberties would be infringed upon and that the government would become tyrannical. The other half believed that a strong federal government was necessary. They solved this by introducing the bill of rights which enumerated how it would protect people and states. These are just some of the problems that existed, the most famous ones.