Could we send someone to a mission into another galaxy

Of course we could try right now. There wouldn't ever be any results in your lifetime, or your children or your children's children. Right now the fastest thing man has created is the Pioneer 10 space probe. It gained a lot of speed using planets gravity to "sling shot it" faster into space. It is going around 11 miles per second. Even if we aimed that at the closest Earth like planet to us it would probably take about 350,000 years to reach.

That's the closest Earth like planet! And there are many many stars in our galaxy very much farther away that would take this space probe even hundreds of thousands of years longer. That being said for the fastest thing man has ever made trying to reach another galaxy : Probably never because the galaxy itself would be speeding away from us. Maybe the Andromeda galaxy (since it is coming toward us) but then even though it's coming straight at us, you would be looking at a couple billion years before the fastest thing man has made to reach it.

And that's without man on-board! lol

Edit: I have a close friend over right now as I was making this answer and he said no, Pioneer is not the fastest that New Horizons probe was. I argued with him then did some research. I guess I stand corrected, although it doesn't change much in the fact that it is impossible to go to another galaxy anytime soon.

Not possible, I'm afraid.

The closest galaxy to ours is the Canis Major Dwarf Galaxy, and is 25000 light years away.
The fastest space-going vessel we have currently built is Voyager I, which is currently traveling at 1/18,000th the speed of light.

For this probe to reach Canis Major Dwarf, taking the most direct route would take 450 million years.

Even a trip to the closest star, Proxima Centuri, is 4.23 light years away, and would take 76140 years to achieve.

Obviously, with modern technology, we could probably improve on the speed of Voyager I, but it would still be several orders of magnitude to slow for even a probe mission to be remotely feasible, let alone a manned one.