Where do comets and asteroids go?

Answer 1

Most "asteroids" are in orbit around the Sun, unless disturbed by gravity or collisions. Comets may orbit the Sun regularly, or be drawn in from the outer solar system and pass by the Sun on their way back out (hyperbolic comets).

Answer 2

Comets and asteroids can have various paths in our solar system. Comets often have elliptical orbits that can take them close to the Sun and then out into the outer solar system. Some comets can even leave the solar system entirely. Asteroids typically have more stable orbits in the asteroid belt between Mars and Jupiter, but they can also be influenced by gravitational forces to move closer or farther from the Sun.