According to the laws of physics, at least as we currently understand them, it is impossible to go faster than the speed of light. That speed limit, in a cosmos as vast as ours, is problematic to say the least.
That speed limit, and whether or not it is obeyed within a science fiction story, determines the kind of story you can tell.
In Star Wars and Star Trek, we see ships moving at many times the speed of light. Though Star Wars uses hyperspace and Star Trek uses warp, the effect is largely the same when it comes to storytelling. The story isn’t inhibited by an inability to travel between the stars. The protagonists in those stories are free to interact with a plethora of alien species, visit a multitude of worlds, and generally get into any kind of trouble the imagination can conceive.
On the other hand, there are examples of sci-fi where the speed of light cannot be surpassed, yet humans still try to travel amongst the stars. Barring the existence of a wormhole, which has often been used even in universes where faster-than-light travel is possible to get around the any limitations those technologies may have, humans have to travel the long and slow way. In scenarios like this, we might see humans go into some kind of long-term sleep, which enables the human players within the story to still make it to their destinations within the allotted amount of time.
However, when there are no shortcuts to be had, and no way to sleep peacefully through the journey, we get generational ships. These are ships where generations of people are born and die before the ship comes anywhere near its destination. They never know a life outside the ship.
I’ve never written a story of this type (though it sounds tempting), but if I did give it a try, I know a few of the questions I’d like to explore. Let’s say there are multiple ships, each heading to its own destination, and there is limited contact between them. How long would it take for the people aboard each ship to develop their own distinctive cultural traditions? How long before the languages changed so much between ships that it would no longer be possible for them to talk to one another? Evolution would also continue on its course. How long before the people between ships differed so much that they could no long breed with one another? How long before they no longer recognized one another as members of the same species?
And, of course, after a group has lived on a spaceship for such an prolonged period of time, would it even be feasible for them to settle on a new world?