What causes the precession cycle or earth? Do other planets experience this type of a cycle?

The Earth experiences several different precession cycles.

One precession cycle is called apsidal precession or perihelion precession. This means that the imaginary line connecting perihelion and aphelion is slowly rotating about the Sun. It takes about 112,000 years for a complete cycle.

The apsidal precession is caused by perturbations from the gravity of other planets. There is also a component caused by the fact that the Sun and the Earth have equatorial bulges which causes tidal effects. There is also a small component due to relativity. In fact general relativity was used to explain a perturbation in the orbit of Mercury.

Another form of precession is the precession of the equinoxes. This is where the Earth's axis of rotation is precessing. It is analogous to the wobble of the axis of a spinning top. It is mainly due to the pull of the Moon and Sun on the Earth's equatorial bulge. It has a cycle of about 26,000 years.

All moons and planets experience such cycles. The effects were first observed in the Moon's orbit over 2000 years ago.

The precession cycle of Earth is primarily caused by gravitational interactions between Earth, the Moon, and the Sun. Other planets also experience precession, but the causes can vary depending on the specific planetary system and gravitational influences from other celestial bodies.