Many things orbit the sun, the most common associated with orbiting the sun are the major planets:
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
There are also asteroids, comets, dwarf planets, etc. that orbit the sun.
The planets orbit (travel) around the Sun.
Everything in the solar system other than the sun itself is in orbit around the sun. This includes planets (such as the one we live on), asteroids, and comets.
icy objects travel around the sun
That doesn't make any sense. If you ment, 1. "Does the sun travel around the earth?" or 2. "Does the earth travel around the sun?" The answer is 2.
the sun does not travel around its self
The sun cannot travel around itself.
They rotate. Travel around the sun is called revolving.
The planets travel around the sun in elliptical orbits due to the sun's gravitational pull. This movement is known as the heliocentric model, with the sun at the center of the solar system, contrary to the geocentric model where the Earth was believed to be at the center.
Many objects travel around the sun, including planets, dwarf planets, asteroids, comets, and artificial satellites. The sun's gravitational pull keeps these objects in orbit around it, following predictable paths.
the moon travels around the earth,and the earth travels around the sun
Everything in our Solar System revolves around the Sun. That is why there is this "Solar System". Because everything revolves around the Sun.
When you travel around the Sun or a planet, due to gravitational attraction, it is called an "orbit". It is also a "revolution" around the Sun or planet.