Trying to identify a book where the Earth is destroyed by aliens, using some probes. A generation ship is sent to find and destroy the aliens.
Edit: more things I remember about the story (this was a recommendation of a friend, I never read the actual book) the ships goes in a vengeance trip but when they find the aliens many years have passed and they are now different, with no memories of what they did to the earth.
This sounds like Greg Bear's Anvil Of Stars, the sequel to The Forge Of God (which I recently proposed as answer to this question).
From Wikipedia:
Volunteers from among survivors of the recently destroyed Earth are sent on a quest by a mysterious race of beings known as "The Benefactors" to find and destroy "The Killers", the civilization responsible for the Earth's destruction.
Note, this wasn't a generation ship though.
This might be "Five Thrillers" by Robert Reed, as per Looking for Sci-Fi Short Story: Avenging unknown alien beings that destroys sun. The "probe" would be the alien ship that pilots into the Sun's corona, causing a reaction that will result in a supernova. And it's not actually generation ships, but rather the farflung bits of humanity who are told to strike back in the ensuing generations.
"It has been an honor to serve as your President,” Joe told an audience of two and then three and then four billion. But most citizens were too busy to watch this unplanned speech—an important element in his gruesome calculations. “But my days are done. The sun has been infiltrated, its hydrogen stolen to use in the manufacture of an amazing bomb, and virtually everybody in the range of my voice will be dead by tomorrow.
"If you are listening to me, listen carefully.
"The only way you will survive in the coming hell is to find those very few people whom you trust most. Do it now. Get to your families, hold hands with your lovers. Whoever you believe will watch your back always. And then you need to search out those who aren't aware of what I am telling you to do.
"Kill those other people.
"Whatever they have of value, take it.
"And store their corpses, if you can. In another week or two, you might relish the extra protein and fat."
He paused, just for a moment.
Then Joe said, “For the next ten generations, you will need to think only about yourselves. Be selfish. Be vicious. Be strong, and do not forget:
"Kindness is a luxury.
"Empathy will be a crippling weakness.
"But in another fifty generations, we can rebuild everything that we have lost here today. I believe that, my friends. Goodness can come again. Decency can flower in any rubble. And in fifty more generations after that, we will reach out to the stars together.
"Keep that thought close tonight, and always.
"One day, we will punish the bastards who did this awful thing to us. But to make that happen, a few of you must find the means to survive!"
Perhaps Forgotten Colony? That's a series name actually. Author is Michael R. Forbes.
The books in it are:
SPOILER ALERT: Everything that follows (till the last couple paragraphs) could easily be considered as giving away far too much. So if someone who is not the poster is reading this, now is a good time to start to scroll slowly, ready to stop quickly...
The idea is an alien species has come to Earth and devastated it. Details in book, but the species is called the Xenotrife. A Marine officer, Caleb Card, has been the most successful unit leader against them, performing mostly extraction missions for a couple years, but the Earth is gone. Turns out what's left of government has created a generation space ship to go elsewhere and reestablish mankind. Or maybe six of them, though Card only knows of the one he ends up at.
His last mission was to extract a doctor and her team from their own technology recovery mission. She's, shall I say, a piece of work, and a continuing villainess throughout the books.
So, Caleb Card's unit is to be security outside the passenger portion of the ship for the 200 years to get to elsewhere, but that goes sour. Things go sour in the ship as well. When they arrive at a star system and planet, it turns out nothing is very democratic anymore in the ship and the current boss is not the man his father and grandfather were. There are enemies on the planet, and guess what? Not coincidentally, thank you old government swine, but the planet has Xenotrife on it. And more.
Shenanigans ensure and eventually everything works out. Works out for the good? For the bad? Hmm...
So:
If none of that rings any bells, this might not be the book/series you have in mind. If any of it does, well, then it is.
The aliens arrived via probes that landed and disgorged the Xenotrife who began killing humans that survived a disease they also brought (engineered for us)... at least I seem to remember a disease to cull our herd. This matches your description. Additionally, as mentioned below, it turns out the generation ship's destination was chosen by figuring the best backtrack for those probes, turning out to be a planet they call Proxima, and the idea turns out to be that the ship's humans were meant to take the war to the enemy. What they find is not what one would expect and they lost track of that mission completely (not that it was broadly known at all) in the uprising they had along the way. This is also a fine fit to your description.
Caveat: The author has written a lot of books and some of them seem, by series name and kind of material in his blurbs on his website, to be from the same conceptual space. Like one set seems to be people in one of those other ships. That one has a "sheriff" named Hayden Duke and the ship is called Pilgrim. If it is related, and "demonic aliens" certainly describes the Xenotrife to a "T" (literally, they were described that way in the above books, first one anyway). If related, probably enough backstory was given to certainly fit your question.
A different series, Forgotten Vengeance, seems to take over after the series described above. The focus seems to be on the "universic" enemy I mentioned as the ultimate bad guys (going up a chain of them...) from that series. Surely the series includes enough backstory to set the stage, so this is the second guess I would make if the Caleb Card series rings no bells. Then (possibly) the Hayden Duke series (just above).
Side note: I "bought" them for $0.00 on Amazon (No idea what price the set currently has on Amazon though.) and certainly got my money's worth and more. However, not too much more... I read them all, don't get me wrong, but except for periodic moments, I felt they slowly became a "here's a horrifying immediate death problem, oh here's the planned solution, rinse, repeat, rinse, repeat, rinse, repeat..." kind of thing. You know, the kind of thing "Young Adult" books do to trade on the constant force majeure, God stepping on the ants, feeling that teens experience as adults admonish them to act like adults, but, oh, not that, and here's something I never even told you about and here's another, and so on. Read them all though.
That said, I have read some of his other works and enjoyed them.
Obviously this is a fairly common trope, as evidenced by the number of viable answers!
I'm going to suggest the Orphans trilogy by Sean Williams and Shane Dix:
Earth is destroyed by alien nano-technology and some surviving humans set off in pursuit of the aliens. It's not quite a generation ship, in that the hunting humans have their memories stored. Still, it's close enough that I'm putting it forward as an answer.
CHRYSALIS. It’s a short story I am listening to
