What is the speed of progression of the sun toward and away from the galactic equator during its orbit?
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. The Sun is currently above the plane and moving upwards, and each cycle takes about 70 million years with an amplitude of 100 pc; it will be roughly 30 million years before we cross the plane again.