Is the Sun in our solar system moving or stationary? When I was small, I read that Sun is fixed at the center of the solar system and that all the other planets rotate around it.
But later I heard that even the Sun is not fixed; it moves. Is this true?
Why had people previously thought that the Sun is fixed?
Is this correct planets in helix path or spring like path?
The Sun moves, even in the context of the solar system. Gravity of the planets (mostly Jupiter) pulls the Sun out of position with respect to the centre of gravity of the solar system. This wikipedia entry explains it in a lot more detail, and also explains that their common centre of gravity lies outside of the sun.

This wobbling of a star due to planets orbiting them is also one method of how we detect planets around other stars.

Assuming that the Sun is fixed within the context of the solar system is a fairly good approximation. You need accurate and long term observations to detect the wobble.