The Sun orbits in the Galactic potential. The motion is complex; it takes about 230 million years to make a circuit with an orbital speed of around 220 km/s, but at the same time it oscillates up and down with respect to the Galactic plane every $\sim 70$ million years and also wobbles in and out every $\sim 150$ million years (this is called epicyclic motion). The spatial amplitudes of these oscillations are around 100 pc vertically and 300 pc in the radial direction inwards and outwards around an average orbital radius (I am unable to locate a precise figure for the latter).
This is established by measuring how the Sun moves with respect to the average motion of the stars in the solar vicinity - the so-called local standard of rest. Estimates of the Sun's motion with respect to the LSR vary a little. According to Dehnen & Binney (1998) the Sun currently moves at 10 km/s inwards, at about 5 km/s faster than the average star tangentially to the Galactic centre and at about 7 km/s upwards out of the Galactic plane. Though as I mentioned previously, these motions are of an oscillatory nature around the mean value. The Sun is currently about 8.5 kpc from the Galactic centre and about 20 pc "north" of the Galactic plane.
I leave it as an exercise for the reader to visualise this orbit. From a side-view it would look like a pseudo-sinusoidal oscillation. From above the disk it would look like a slightly eccentric ellipse with a very large precession.
All of the above assumes that the Sun moves in a smooth Galactic potential. In fact the presence of spiral arms, a Galactic bar and the clumpy presence of giant molecular clouds can and will perturb the Sun's Galactic orbit around the regular behaviour described above, so predicting the Galactic location of the Sun in a billion years or so is not necessarily going to be very accurate.