The first thing that would happen is Earth would switch roles. We’d become the moon of Saturn. Saturn is almost one hundred times as massive as Earth. It’s even bigger compared to Earth than Earth is to our moon.
By the time Saturn is at the same distance as the Moon, its tides would be many thousands of times stronger than the Moon's. Fault lines would rupture, volcanoes would blow their tops, and anything left on the surface of the Earth would be wiped out.
Second, the end results of being Saturn’s moon would depend on what orbit we ended up in. Saturn has a lot of moons and rings that Earth could collide with. Eventually, Earth would also end up tidally locked to face Saturn, which would make our days very long. Overall, it’s not a happy situation for Earth.
In the long run, there’s also the problem that having a giant planet like Saturn in the inner solar system would disrupt the orbits of Venus, Mars, and Mercury.
Rather than dooming Earth by moving it into Saturn’s orbit, a more fun thought experiment might be to consider existing habitable moons orbiting gas giants in other solar systems.