There is a general misconception that we cannot travel faster than light or reach distant stars or even galaxies in our own lifetime. The problem is with the word WE. There is no we. People back on Earth and people on an interstellar spaceship see things quite differently.
The astronauts certainly can in principle reach galaxies millions of light years away in their own lifetimes but the people back on Earth will have to wait millions of years to find out about it.
The motion part of the special theory of relativity which Einstein published in 1905 stands or falls on the truth of the Principle of Relativity. This states that there is no absolute velocity and there is no experiment that passengers in a vehicle that is moving at fixed speed in a straight line can perform to measure their speed and even tell that they are moving. They could look outside but all they would discover is that they are moving relative to their surroundings.
The dynamical equations of Special Relativity, which I might add later are little more than a complicated form of Pythagoras’ Theorem in four dimensions, because time is now a dimension. They tell us what a person standing on Earth will record if they try to measure lengths and look at clocks on a moving vehicle. There is also an equation which tells us what the Earthling will see as the mass of the spaceship, which will generally be greater and even infinite if the ship is travelling at the speed of light.
You will read, even in the works of celebrated authors, that spaceships cannot reach the speed of light because their mass would approach infinity and so an infinite amount of fuel would be needed. That might be true if by some miraculous means the fuel was being supplied from Earth.
For our brave cosmonauts there would be no such effect since acceleration is absolute, not relative, and so they could see how much fuel was needed to provide a certain degree of acceleration. If they could show the mass of the ship was indeed increasing, then they could use this to compute an absolute velocity. And the entire theory vanishes in a puff of smoke.
Now what about the journey itself. It would take a few decades because for the comfort of the crew the acceleration would need to be around the same as the acceleration due to gravity of Earth. Once close to the speed of light almost all the journey would take place in a very short time, no matter what the distance travelled. Some very precise navigation would be needed as the ship would have to turn around and spend a few more decades decelerating. So our space travelers could reach distant worlds in their own lifetimes but there would be no going home to the same world they left. It might not even exist.
Now for the slightly harder bit. The version of Pythagoras’ Theorem which applies to travel in four dimensional spacetime is called The Lorentzian Pseudo Metric. This gives the time measured by an ideal clock which travels a certain distance in a certain time. But unlike the simple Pythagoras’ Theorem that we all know and love, this one contains minus signs. When the distance part equals the time part then clock measures zero time. Go any faster and the time becomes negative which is not allowed. So the speed of light as measured by outside distances using its own clock in its own reference frame is actually infinite. This is the cosmic speed limit and applies to all travelers anywhere in the universe. Everyone will measure it as 300,000 km per second.
The universe has a speed limit, the same everywhere to avoid violation of causality. Sending messages into the past is not allowed.