Given enough time, where are the Voyager spacecrafts heading? (Assuming some alien civilization doesn't pick them up.)
Will they pass by any interesting stars on the way to the black hole at the center of our galaxy or will it perhaps leave the galaxy?
What are the highlights on their journey that we can reasonably predict?
Neither Voyager 1 nor Voyager 2 was aimed at any particular target outside the Solar System. Their trajectories were largely determined by the requirement to do fly-bys of Jupiter and Saturn (and, in the case of Voyager 2's extended mission, Uranus and Neptune). They've been sending back some interesting results about the boundary between the Solar System and interstellar space, but are not expected to remain operational long enough to send back any data about any other stars.
Voyager 1 happens to be heading "in the general direction of the Solar Apex (the direction of the Sun's motion relative to nearby stars)", which means it should reach the heliopause somewhat sooner than if it were going in a different direction. I don't think that was deliberate. To do that intentionally, they would have had to alter its course as it passed through the Saturn system; as far as I know, its course was optimized for observations of Saturn, its rings, and its moons. And Voyager 2 passed through the termination shock 10 AUs closer to the Sun than Voyager 1 did.
I don't think there's any particular reason to think that either one of them will encounter the black hole at the core of the galaxy. To reach the galactic core, they'd have to have enough velocity to cancel the Sun's orbital motion around the core, about 251 kilometers per second, compared to their actual Sun-relative speed of about 17 kilometers per second.
JPL's web site for the Voyager Interstellar Mission says the following:
Both Voyagers are headed towards the outer boundary of the solar system in search of the heliopause, the region where the Sun's influence wanes and the beginning of interstellar space can be sensed. The heliopause has never been reached by any spacecraft; the Voyagers may be the first to pass through this region, which is thought to exist somewhere from 8 to 14 billion miles from the Sun. This is where the million-mile-per-hour solar winds slows to about 250,000 miles per hour—the first indication that the wind is nearing the heliopause. The Voyagers should cross the heliopause 10 to 20 years after reaching the termination shock. The Voyagers have enough electrical power and thruster fuel to operate at least until 2020. By that time, Voyager 1 will be 12.4 billion miles (19.9 billion KM) from the Sun and Voyager 2 will be 10.5 billion miles (16.9 billion KM) away. Eventually, the Voyagers will pass other stars. In about 40,000 years, Voyager 1 will drift within 1.6 light years (9.3 trillion miles) of AC+79 3888, a star in the constellation of Camelopardalis. In some 296,000 years, Voyager 2 will pass 4.3 light years (25 trillion miles) from Sirius, the brightest star in the sky . The Voyagers are destined—perhaps eternally—to wander the Milky Way.
Both Voyagers are, of course, still influenced by the Sun's gravity. Ignoring other forces (which is a good enough approximation for thousands of years), they'll continue to decelerate, asymptotically approaching a speed that depends on the Sun's gravity and their kinetic energy. According to a footnote in this article,
Voyager 1 has an asymptotic velocity of 3.5 AU/yr, Voyager 2 an asymptotic velocity of 3.4 AU/yr
which converts to about 16.6 and 16.1 kilometers per second. That's not much slower than their current velocities.
Reference (Wikipedia footnote): Mallove, Eugene F.; Gregory L. Matloff (1989). The starflight handbook: a pioneer's guide to interstellar travel
So basically voyager 2 is not really moving at all. Our solar system is traveling away from the center of the galaxy at approximately 35,000 miles per hour and voyager is traveling toward the center of the galaxy or away from the our sun at the same speed.
