Other galaxies are really far away, so that it takes millions of years for even light to get there. Therefore I don’t think we’ll get there anytime soon. In principle we could send a probe to study Andromeda, but even if we could make it fast at the speed of light, we would have to wait five million years for the results.
However, because time slows down when moving really fast, the travel time would be much shorter for an observer in the probe. Therefore, if we can develop the technology to reach speeds like that, then someone could actually try it. I think that eventually it will happen, but in a far future.
Alternatively we could try to change the motion of a star and send it towards Andromeda, together with a planet with people living on it. It would take a large number of generations, but eventually they would get there.
I hope we do eventually get to explore beyond the solar system. A few million years (which is what it would take to spread out over the galaxy) is not actually that long compared to the age of the Universe, or even the length of time that life has been on Earth. You could certainly imagine that robotic probes sent out from Earth could cover the galaxy eventually. It’s hard to imagine our current politicians thinking about a project that will take millions of years though (they barely seem able to think 4 years into the future), but technologically speaking we’re not far short of being able to send such probes out into the galaxy.
Another answer to this question is that it is thought the Milky Way will collide with the Andromeda galaxy at some point in the far future (they are falling into one another under gravity).