One of the biggest science fiction problems to solve is the speed of light. Everyone knows you can’t go faster than it; everyone also knows there’s not many places you can get to going slower than it. But beyond that, it gets confusing. How close to the speed of light can you go? What happens when you do?

How the cosmic cops pull you over
When you approach the speed of light, you don’t blow up or anything. Instead, your mass increases and you stop speeding up as fast as you did. But from your perspective, time starts to dilate. So you feel you are continuing to accelerate, but since time is going slower for you, you’re actually accelerating less and less. You don’t stop accelerating, but instead of going from 99% c to 100% c, you go from 99% c to 99.9%, to 99.99%, to 99.999%, and so on.

What this means, effectively, is that if you could get close enough to the speed of light, you could travel anywhere in the galaxy in your lifespan. Only problem is, thousands of years would have passed on Earth. That can be very interesting in a science fiction plotline.
Light, by the way, can’t go faster than the speed of light either. That means information can’t. So if you do make it across the galaxy, you can’t call home without some more science fictional tech.
Ways to get away with speeding
The simplest and, in my opinion, most plausible way to go faster than light is by some kind of wormhole or hyperspace. If you could go through the fabric of spacetime instead of across it, you wouldn’t have to go faster than light to get places quickly.
I love reading about different ways this works in different SF universes. For instance, if you have permanent artificial wormholes, some kind of “gate,” then that affects the strategies people can use. Blockading gates could be helpful. The location of the gate–near a planet, near the star, out by the outer planets–becomes crucial, and affects how long it will people to get from planet to planet. (Kind of like how far you live from the airport can be as important to your travel time as the distance you’re traveling by air.) Or do the gates move? Since stars and planets move, I’ve always liked the idea of a planet eventually getting out of range of a natural wormhole and having to find another one.
Alternatively, ships could have some kind of spacetime cutting device to create their own wormholes. Star Wars ships get into hyperspace all on their own (though there are rules about when you can make the jump). Then you can ask questions like “How long do they spend in hyperspace” and “Can they travel the whole distance in one go, or do they have to make multiple jumps?” Or what if it’s deeply weird in hyperspace and will drive men mad?
One of my favorites is Zahn’s Quadrail series, where a spacetime tunnel is converted into a literal railway and you take trains to other planets. TRAINS IN SPACE.
Then again, you could always just take the Star Trek approach and have special engines that go faster than light in normal space. In that case, they use some kind of warp bubble and contort spacetime so that they’re never actually going faster than light. I don’t claim to understand the Alcubierre Drive, but that’s the concept of it.
Sublight fun
But you don’t have to have a faster-than-light travel device to have fun in space. You can, as I mentioned earlier, travel at relativistic speeds and just deal with everyone back home growing old and dying while you’re on your way.
Or you can stick within the solar system. The Expanse showed you can travel at a pretty good clip all over the solar system, if the speed of light and the g-forces you pull are your only limitation. In fact, my main complaint about those books is that they don’t make it clear just how fast our heroes are going. If you can accelerate at 1g and then flip and burn, you can get from Earth to Mars in a matter of days. To Neptune in a month!
In fact, if you could accelerate at 1g indefinitely (which, in The Expanse, you can’t because of nebulous fuel concerns) you would reach very near lightspeed in about six months. Therefore you could get from Earth to Proxima Centauri in about five years. That’s really quite doable!
…if, as I said, that was your only limitation. Currently, some kind of rocket is the main way we travel in space, and it requires constantly throwing mass away. But the more fuel you carry, the heavier your ship is and then the more fuel it takes to move. It starts to be a problem long before you’re jaunting from Earth to Mars over the weekend, let alone Alpha Centauri. Hence The Expanse’s magic, unexplained engines which can burn indefinitely, unless fuel efficiency is suddenly needed for plot reasons.
We can, of course, theorize other methods of travel. Solar sails are one that we know works, though they have their own limitations. To get much thrust, you need an enormous sail, and sails have mass that also needs thrust to move. It’s similar to the rocket problem. You can go indefinitely without much trouble, but if you want to go fast, the sails need to be improbably huge.
But I can imagine other kinds of engines that could work. Perhaps if we discovered another kind of matter, previously invisible to us, that we can push off of while in space. Perhaps the interstellar medium is full of enough stuff that we can scoop it in the front and push it out the back. Scientists are always searching for a way around this problem, and it’s not a hard limit the way lightspeed is. It’s very plausible that they eventually solve it.
Once you understand the barrier lightspeed is to casual jaunts around the galaxy, your imagination is the only limit to how you get around it. My only pet peeve is people who don’t understand it messing things up. Like that bit in The Expanse where we are told you can’t get too close to the speed of light or else “Einstein gets cranky.” Uh no, time slows down and you don’t make as much progress as you’d think, but nothing bad will happen. And charging around at accelerations of 1g or more, our characters go an appreciable fraction of the speed of light all the time. (This was retconned later as “fuel concerns.”)
You can calculate time dilation with this handy widget and be Einstein-correct in everything you create, if you care to!