The rocket equation tells us that if one somehow managed to build a spacecraft containing an infinite power source that weighed a single kilogram, and incorporated the most efficient engine designs known to us (electrostatic ion thrusters), it would still need to be fueled with a reaction mass of xenon equivalent to approximately two billion times the estimated mass of the observable universe to achieve even 10% of the speed of light.
For example, if you had a vehicle weighing 10t empty (but including energy), and a thruster capable of accelerating ions to 1% the speed of light, you'll only need 17t of reaction mass to achieve a speed delta of 1% the speed of light.
This ratio holds up as you scale exhaust velocity and delta V, so the exact same reaction mass would be needed for a 10% light speed delta V using a thruster shooting ions at 10% c. As a rule of thumb, you always need a reaction mass about 170% of the vehicle's dry mass when your thruster exactly matches your target velocity.
Now, to illustrate how critical exhaust velocity is, imagine you make thrusters with exhaust velocities 10x that of the target delta V. In those cases, you'll only need a reaction mass of roughly 10%. However, if your exhaust velocity is 0.1x the target delta V, you'll need a 2200000% reaction mass!
1: https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation 2: https://en.wikipedia.org/wiki/Ion_thruster#Comparisons 3: https://en.wikipedia.org/wiki/Observable_universe
Not that the former is easy, but the latter is basically impossible. After the former is achieved, we can basically coast on sleep for eternities.
Some time after the ship gets launched (centuries, even), FTL gets developed. Then they either arrive and find a civilization at their destination, or they are considered to be off-limits and not to be messed with, depending on the setting.
That assumes such improvements are even possible.
At least for non-magical propulsion, we don't really need "faster" engines. Take whatever engine you want, let's say it can accelerate at a constant 1g. In less than 4 years(from an observer standpoint), you are at 97% of lightspeed. From your point of view, that would take less than two years.
If you can have an engine that can accelerate at a constant 1g for years (fueling it is left as an exercise for the reader), then you are all set. You can go anywhere in the galaxy inside a human lifespan – of course, assuming you don't care for the people left behind.
If you want to accelerate at more than 1g, there may be physiological issues.