I know that the farthest man-made object from Earth is Voyager 1, and was wondering if and when it will ever come relatively close to another star system, or some other object located outside the Solar System? If this will happen, when, and will it come close enough where it will begin orbiting the object, for example a star or planet?
I don't know what lies along either of the Voyagers' flightpaths (and would be skeptical of someone that claimed they did), but I am pretty sure that they will have escape velocity for any solar system or object they pass.
Why? Because escape velocity is the velocity where the object's kinetic energy and potential energy relative to another body is the same as a stationary object an infinite distance from that body. Since the Voyagers are a very very large (effectively infinite) distance from any body they might orbit and have a non-zero speed, they will have escape velocity for that other body.
The only way this might change is if there was some significant but non-destructive drag imposed on the spacecraft as it approached another object. It is more likely that any such drag would tear the spacecraft apart.
There are no major objects that either Voyager spacecraft is planning on reaching anytime soon in the Solar System. The best that I could find after considerable digging around is that Voyager 1 will come 1.6 light years from the star designated AC+79 3888 in about 40,000 years, sourced here.
There is always the oddball chance that one of them will run in to an Oort cloud object, but we can't even see any of them from Earth, so that's quite difficult to determine. For a rough pass of the two spacecraft, see this animation.
By far, the best science that they can do is to learn more about the heliosphere, which they were designed to do and have power to do. And as it surrounds the Earth, they will run in to it from any direction.