We can't travel beyond the edge of the observable universe. That space is expanding away from us faster than the speed of light.

There's only three conceivable ways that we could ever travel outside our local group

1. we create some generation ship that travels at a significant fraction of the speed of light. It would take 50 or 100 million years for that ship to travel to the Vergo Cluster of Galaxies, which is the next closest large galaxy group to us.

2. we create some faster than light warp drive mechanism. Recent calculations have suggested that it might be possible if we could convert the entire mass of Jupiter into energy. For this we would be limited by the speed the warp drive is able to provide. If it was even some single digit multiple of the speed of light, it would still take 5 to 10 million years to reach the Vergo Cluster.

3. we create a worm hole that allows us to travel any distance very quickly. This would also require negative energy or negative mass to hold the wormhole open. We don't have a theory of quantum gravity that could even tell us if this is possible, but most current theories say that any wormhole would collapse immediately, or not allow mass (or possibly even light) to travel across it.

Keep in mind that due to relativistic time dilation, traveling faster than light is basically the same as traveling back in time. Also wormholes could theoretically be manipulated to create time portals that allow forward and backward movement in time.

Anyways, the Vergo Cluster is loosely gravitationally bound to our own local group. If you wanted to travel to a galaxy that was not gravitationally bound to us in some way, that might take 50 to 500 million years of travel (depending if faster than light travel is possible), mostly through voids of interstellar space. That would require taking the expansion of the universe into account for trajectory calculations.