Sun’s gravity well is so immense that escaping it requires a very, very large amount of energy. I am not sure if it could be done with conventional rockets with current technology, without multiple gravity assists.
Edit: As the child comment says, it seems to be possible with current technology. It is very exciting!
Yes, it can. New Horizons is escaping the solar system only a little bit slower than the Voyager probes, and it was already on an escape trajectory before its single Jupiter gravity assist.
The Voyagers didn't have an extrasolar target either, they were meant to visit the outer planets. If we're looking at probes that leave the solar system after their original mission, we launched New Horizons in 2006 to Pluto, and it's currently in the Kuiper Belt on its way to leave the solar system.
Meanwhile, there are actual interstellar probes being planned, in addition to several missions to the outer solar system (and which could end up in interstellar space afterwards).
And there's a ton of missions planned for the inner regions of the Solar system -- space exploration is still a thing and there's a lot of exciting things happening.
Through (according to Wikipedia) the party might end in just a few years: https://en.wikipedia.org/wiki/Voyager_1:
> Voyager 1's extended mission is expected to continue until about 2025, when its radioisotope thermoelectric generators (RTGs) will no longer supply enough electric power to operate its scientific instruments.