The verbs are all over the place in this example, so I’m going to make it all past-tense.
I’m also changing spaceship to starship, because starship seems more appropriate for something with interstellar capability rather than merely the ability to travel to Earth’s moon or some other location within Sol’s system.
(I have a problem with the use of the word elite in this context. One way or another, it doesn’t work.)
When the linking verb is the same for both parts of a compound predicate, you can omit the second use of the linking verb. In this example, that means you can omit was (formerly is) before launched.
I recommend deleting off and reach as unnecessary words.
Changed habited to inhabited. (Habitable planets can be inhabited, but nuns are habited: wearing or dressed in a habit.)
Humanity is an awkward word here; change it to humans.
The they in the last sentence is unclear and ought to be replaced. Try the ship or the people on the ship.
Write lightspeed as one word to avoid confusion. (Some online dictionaries have it as light speed, but others have it as lightspeed, and it doesn’t appear at all in the print edition of the 2002 Webster’s Universal Dictionary I have on my desk, which seems like a strange oversight on someone’s part.) Also, since it’s used here as an adjective, it ought to be one word (or hyphenated) even if it were otherwise written as two words.
I’ve changed traveled to was en route so there’s no “repeat” with travel later in the sentence.
A generation starship was filled with the chosen elite and launched to a far-away, habitable planet. Upon arrival, their descendants found their new home already inhabited. While the ship was en route, the humans left behind discovered lightspeed travel and got there first.