Where is Voyager 1 (or 2) going? In which direction? Knowing it has exited the solar system, is it traveling in the direction of the center of the galaxy, away from it, orbiting it, or maybe even going perpendicular from the disk-shaped galaxy?
Stars and gases at a wide range of distances from the Galactic center orbit at approximately 220 kilometers per second.


From Wikipedia

That's much faster than the Voyager probes relative to the Sun (Voyager 1: 17 km/s). Hence the probes will orbit the galactic center roughly the same way as our Solar System, even after occasional hyperbolic encounters with other stars.

Voyager 1 is travalling in the direction of Ophiuchus, seen from Earth. She's at or beyond the border of the heliosphere.