Several issue arise, most notably the fact that the faster you travel, the more power you need. Then there’s Einstein’s theory of relativity – For people left behind, time progresses normally, while for the people on a “fast” ship, time moves…at a different pace. If they’re traveling say for 5 years, when they return to Earth, depending on how fast they were going, time may have progressed 10, 20, 50, 100 years or more.
Some people are toying with the idea of “warping” or curving the space near the ship in kind of a wave pattern, with the ship riding the crest of the wave.
I was disappointed several years ago when an experiment with a solar sail failed (the sail never opened). Such an implement could speed up space travel considerably. Another idea is the use of ion engines. I say, why not use both? Point the ion engine at the solar sail, and cruise the galaxy.
We’ll probably get there some day, but not in our lifetimes.