Ah, Trunks. Son of Vegeta and...Bulma o_0.To make a long story incredibly short, thought it still is V. long, after Goku defeats Freiza on Namek, both Goku and Freiza survive and head to Earth, Goku is supposed to kill Freiza once they both arrive and then later die of a heart virus.
Some years later, two androids are supposed to appear and kill every character in DBZ except Gohan, Trunks, Bulma, Master Roshi, Puar, and Oolong. Piccolo was the first to try and stop the androids, and the first to die, upon his death Gohan became a Super-Saiyan.
Some years afterwards Gohan and Trunks try to stop the androids and nearly get killed and Gohan loses an arm. Trunks asks that Gohan train him to become a Super-Saiyan like him. When the androids attack another city Gohan and Trunks go after them, but before they do, Gohan knocks Trunks out and says something really cheesy like "Stay alive so you can defeat the androids one day, I know you can." even though Trunks can't hear him. So Gohan goes and tries to destroy the androids and gets killed. When Trunks wakes up, he goes to the city where the androids were attacking and finds Gohan dead on the floor, he refuses to believe Gohan is dead and keeps trying to wake him up. He then says something really cheesy like, "You were everything to me!" and he shakes his head and starts screaming and becomes a Super-Saiyan.
He then spends the next 5 years training to defeat the androids, and when he hears that they are attacking again he tells his mom, Bulma, that he's going to go stop them. Bulma then says that he isn't any stronger than Gohan was when he died. Trunks fights the androids and soon runs out of energy and finds out the androids never run out of energy and gets beat up REALLY bad and nearly gets killed. When he wakes up again, he's in his house and Bulma is at his side. He then gets medicine to treat the heart virus Goku had and time travels to the time when Freiza arrives on Earth (Freiza arrives less than an hour before Goku does) and Trunks kills Freiza's henchmen, Freiza, and his dad, King Cold.
When Goku arrives, Trunks and Goku fight for fun (Trunks says its to make sure it is Goku, though Goku is the only Super-Saiyan from that time period...). Trunks then explains everything to Goku about his heart virus and the androids, he even tells them the exact place, date, and time the androids will show up, and returns to his own time.
The Z warriors then train for about 5 years until the androids show up, and Trunks returns to their time to check up on things and finds that the androids that showed up in this time aren't the same androids from his time and that one of the androids was already destroyed by Vegeta (he's now a Super-Saiyan).
So he flys torward the battle field and tells everyone the androids aren't the same and also finds that the heart virus has started attacking Goku's heart. Meanwhile, the other android they were fighting takes off torward his laboratory to activate androids 17 and 18 (the ones from Trunk's time).
Vegeta, now thinking he's so strong, wants to wait till 17 and 18 are activated to destroy them. Piccolo and Trunks think its best to hurry and destroy them before they are activated and Trunks tells Vegeta that he has no idea how powerful these androids are. Then Vegeta flies off to follow Dr. Gero (the other android) and Trunks follows him, while they are flying, Vegeta asks Trunks how long he will follow him, and Trunks just keeps warning about the androids' power.
They finally arrive at the cave and Piccolo and the others are already there and they explain that they had already tried to open the door, so Vegeta says, "Stand aside!" and shoots the door with an energy blast that opens it up. Inside they find that 17 and 18 have been activated and Trunks is dead scared. Vegeta just makes fun of the androids.
18 then walks over to a capsule that contains android 16, she says she wants to know what he looks like, and Dr. Gero tells her not to touch it so 17 decapitates him with one kick to the head and smashes it.
Trunks realizes that they don't stand a chance against the androids if they have 16 with them too, so he powers up to Super-Saiyan and destroys the lab, but all three androids were abe to escape.
When 18 activates 16, 17 and 18 try to talk to him but find out he wont talk to anyone unless its about his mission, to destroy Goku, which he was programmed to do. So 17 and 18 say it's about time they carried out their mission to destroy Goku and 16 follows them.
Krillin then says that they have gone after Goku, but Vegeta is saying that he's mad that the androids ignored him and that he's going after them. Trunks isn't a Super-Saiyan at the time and Vegeta is, Trunks tells Vegeta to wait for Goku, Vegeta responds by punching Trunks in the stomach and takes off to follow the androids.
Trunks and the others follow Vegeta and when they get there Vegeta is beat up pretty bad and says he's allright when asked if he's ok. He also says he needs help from nobody.17 then says that if anyone interferes, he will too.
The fight continues and Vegeta is getting pummeled. After a while, 18 kicks Vegeta in the stomach and arm, dislocating it.
Trunks can't take it anymore and jumps in to save him, but gets his sword chipped and his butt kicked by 17. Piccolo and Tien join in and they get whooped too.
Vegeta then gets up and tries to attack 18 again but is thrown to Trunks, knocking them both unconscious.
The androids then leave, blah blah blah. Gohan, Bulma, and Trunks find Cell's exoskeleton thing and Piccolo fights with Cell. Goku gets all better and finds Vegeta and tells him about they Hyperbolic Time Chamber and Vegeta goes to train and Trunks goes with him.
They come out and they go find Cell, Vegeta fights with him and shows that he is far stronger. Cell then tricks Vegeta into letting him absorb android 18. Trunks tries to stop him but is stopped by Vegeta, Trunks then shoots Vegeta with an energy wave and goes after Cell. Cell does a Solar Flare, blinds Trunks, and absorbs 18.
The fight between Cell and Vegeta resumes, and Cell beats the living crap out of Vegeta. Trunks then powers up to the max because he didn't want to do that while Vegeta was still around because he did not want Vegeta to see that he is stronger than Vegeta. Trunks is more powerful than Cell; however being so bulky, he is too slow.
Cell then explains to Trunks that all he did was bulk up and slow down, he can do that too, instead of destroying Trunks, he is interested in how strong he got so fast. He asks Trunks that if he gave him more time could he get stronger. Trunks answers maybe. So Cell decides to hold a tournament and explains the details on they telly.
So the day of the tournament arrives and when Gohan is fighting Cell, the Cell Jr.s attack the Z warriors. Trunks and the others can't destroy the stupid little things, blah blah blah (read Gohan for info on how the Cell Jr.s die). When Cell self-destructs, he regenerates and kills Trunks as soon as he comes back, or so we think. They later find out Trunks isn't dead. When Cell is finally destroyed, they wish everything back to normal with help from the dragon and Trunks finally gets respect from Vegeta and he goes back into his own time.
When he does, he destroys the two androids with the greatest of ease, and Cell as well.
Forms acheived by Trunks and how:
Super-Saiyan: In the original time-line, when Gohan is killed, Trunks finally transforms.
Ultra Super-Saiyan: while training in the Hyperbolic Time Chamber.