And go where?
Not to rain on that parade, but for humans to travel however far a hospitable system might be, they may as well stay there as for how well they'd even be remembered, if they ever returned.
I didn’t say manned spaceships. In truth, the purpose of leaving the solar system is the journey, not the destination.
To start with, it is an endurance contest for engines, most likely ion drive, which would launch in space, not from Earth, and begin to slowly accelerate the further it went. It would reach speeds far greater than from a chemical reaction drive. A fusion drive is also a possibility.
The “edge” of the solar system, where the solar wind meets deep space, is a relatively violent place of great scientific interest. Passing through that area would be the first major test of its detection equipment.
Finally passing into deep space, away from the complex gravitational, energetic and particulate dense solar system would be our first real glimpse at deep space, where a lot of theories could finally be tested.
At this distance, the spaceship also becomes much like the Hubble telescope, and it can act in concert with other Earth and space-based telescopes in several ways. It inherently performs several experiments, just based on its location, such as testing relativistic laws.
One of the more intriguing possibilities will be to test if some quantum phenomena can defy causality, which is unlikely, of if they can be used to transmit data, which might be possible, but difficult.