What is the direction of the movement of the solar system in relation to the galaxy's plane

I think your question is answered by the duplicate mentioned: but here are the relevant highlights.

Humphreys & Larsen (1995) suggest, using star count information, a distance of $20.5 pm 3.5$ pc above the Galactic plane; consistent with, but more precise than the Bahcall paper referred to by Schleis. Joshi (2007) is more guarded, investigating some systematic uncertainties in the estimation techniques and ends up with distances between 13 and 20 pc above the plane.

The Sun moves at about 15-20 km/s with respect to a local standard of rest defined by the general motion of stars in our vicinity around the Galaxy. In three-dimensions, this "peculiar velocity" is $U=10.00 pm 0.36$ km/s (radially inwards), $V=5.25 pm 0.62$ km/s (in the direction of Galactic rotation) and $W=7.17 pm 0.38$ km/s (up and out of the plane). (Dehnen & Binney 1998). Different authors arrive at velocities that differ by $sim 1-2$ km/s from these values and so this would probably be a more conservative estimate of the uncertainties.

The Sun executes oscillations around its mean orbit in the Galaxy, periodically crossing the Galactic plane. I borrowed this illustration (not to scale!) from http://www.visioninconsciousness.org/Science_B08.htm to show this oscillatory motion. The oscillations will not be exactly sinusoidal because the restoring force towards the plane does not very linearly with height above the plane.
As the Sun is currently above the plane and moving upwards, and each cycle takes about 70 million years with an amplitude of 100pc (Matese et al. 1995), it will be roughly 30 million years before we cross the plane again.

EDIT: Actually I'm glad I revisited this question because I think the picture is not very good at all. As the Sun takes ~230 million years to go around the Gaaxy, it should only execute 3 complete vertical oscillations in our Galactic year, whereas the picture implies many more. Secondly, the Sun executes a radial oscillation with a period of around 160 million years, which is not even indicated!