What types of propulsion systems would be needed for intergalactic space travel to become feasible?
At our current level of technology we simply don't know. In fact it may not even be possible other than using "generation shops".
Einstein's theory of relativity states that nothing can travel faster than the speed of light. In fact for any object with mass it requires an infinite amount of energy to reach the speed of light.
The speed of light is so relatively slow that it takes light 2.5 million years to reach the Andromeda galaxy. It would take a space ship a lot longer.
One option is using generation ships which travel at near the speed of light. It would take generations of passengers to reach there and more than 2.5 million years would have elapsed on Earth if and when they arrive.
The other problem is the fuel required to reach near light speed and slow down at the other end. We don;t have the technology for that yet.
Relativity does allow for worm holes as short cuts through spacetime. Unfortunately this requires negative energy material to hold the mouth of the worm hole open. We don't know if negative energy matter exists.
If we are to travel between galaxies we need new physics. We may or may not discover this...