Dragon Ball is the first, and features a young boy named Goku who has extraordinary fighting strength, and his adventures saving the world. The series starts with a very serialised, drama type approach with a rotating villain of the week and an overarching plot of them trying to find the dragon balls, which summon a wish granting dragon when gathered into one place. There’s an evil emperor called Pilaf who also wants to get the dragon balls. at the start of the series, goku is an orphan – his adoptive grandfather, gohan, died shortly before the show started. on this first adventure Goku meets Bulma, a young girl who’s good with technology and the heiress to a massive manufacturing company called capsule corp who make capsules that can summon basically anything (cars, bikes, houses, desks, big bags etc). she takes him to meet Master Roshi on turtle island, a martial arts trainer who can help Goku hone his skills. While there he meets Krillin, Roshi’s other student. they are initially rivals, but soon become lifelong best friends. they have many adventures searching for the dragon balls, and meet other friends like oolong, a shape shifting pig, and yamcha, a rebellious teen who they befriend. basically every person who becomes friends with goku starts out as a villain. the first few arcs revolve around the World Martial Arts tournament, where they also meet Tien and Chao Tsu, who start as villains but join the gang later. it turns out that Goku can turn into a giant, super powerful ape when exposed to the light of the full moon, and it’s theorised that he actually killed his grandfather (Goku only knows it was a giant monster, he woke up one day and their house was destroy and Gohan was dead). the series continues with several sagas, involving the Red Ribbon army who are evil and want the dragon balls, and the King Piccolo saga, where you find out who made the dragon balls: an alien called Kami who is guardian of earth. long ago Kami split himself into good and bad parts, the bad part became King Piccolo who tries to invade earth but fails (thanks again to Goku). At the end of the series Goku marries a girl called Chi Chi who he promised to marry way back at the start of the show (her dad started as a villain and, you guessed it, becomes a good guy). oh, also, Master Roshi teaches Goku the Kamehameha wave (pronounced ka-MAY-ha-MAY-ha wave, rather than KA-mee-HA-mee-HA), which is a super powerful beam attack.
Dragon Ball Z takes place a few years later, when Goku and Chi Chi are grown up and have a kid of their own, Gohan, named for goku’s grand father. The first arc of this series is the Saiyan saga, where an alien named Radditz comes to earth and claims to be Goku’s brother, and says his real name is Kakkarot. Radditz says Goku is really a Saiyan, an alien race of super powerful warriors who destroy planets for profit. Goku was sent to earth to destroy all intelligent life on the planet to make it ready for sale, but he bumped his head as a baby soon after Gohan Senior found him, and his personality changed completely. Radditz goes on to say that all the Saiyan’s have been killed except for the two of them and a handful of others. Goku doesn’t believe Radditz to begin with, but if you think about it makes a lot of sense. Obviously they fight because a) it’s dragon ball z and b) Goku doesn’t want to be a planet destroying warrior. The heroes gain a pyrrhic victory where King Piccolo’s son/clone/thing (also called Piccolo) kind of becomes good, and kills Radditz and Goku at the same time (at Goku’s request because there’s no other way to beat him). As Radditz dies, he says there are other Saiyans who will come, who are more powerful, and will destroy them all. Obviously the gang (who at this point is Krillin, Yamcha, Tien, Chao Tsu, Piccolo and Bulma) are pooping their pants because Goku, their best fighter is dead. But Goku is in the Other World, and goes to train with this super master guy called King Kai while the gang gather the dragon balls to wish him back to life. While Goku is dead, Piccolo trains Goku’s son, Gohan, in the ways of the warrior, because he too shows incredible fighting potential. The other Saiyans are Napa and Vegeta, who eventually show up and wreak havoc. The gang hold them off until Goku comes back (it’s this arrival that prompts the classic “It’s over 9000!” quote). Through another series of narrow victories (which includes Vegeta doing the giant ape thing and getting his tail cut off by some low life nobody named Yajirobe, come at me fanboys), they defeat the Saiyans, killing Napa but only severely injuring Vegeta. Goku allows vegeta to escape alive, stating that Piccolo became good, so who knows with Vegeta.
Vegeta runs away, and thus begins the Frieza arc. Vegeta killed a bunch of people, including Piccolo, so for reasons (just go with it) the dragon balls disappear, so they have to travel to Kami’s home world to use the dragon balls there to wish everyone back. On the planet (called Namek) they meet Frieza, a super powerful, super ruthless alien thing that can basically pwn anyone he wants all day long (he’s the one that killed all the saiyans). He also wants the dragon balls to become immortal. After an incredibly long, incredibly drawn out battle which includes Vegeta dying and seemingly becoming good on his death bed, Goku’s true inner potential as a warrior god becomes unlocked, and he becomes a Super Saiyan (i’m sure you’ve seen the pictures of the blondie – that’s how Goku looks in super saiyan mode). Frieza eventually becomes out classed by Goku’s raw power and is defeated, but destroys the planet Namek as a last sort of half trump card half hissy fit type move. Everyone escapes (except Goku, who stays behind to finish frieza), and in the mean time they manage to use Namek’s dragon balls to wish everyone Frieza killed back to life. Goku seemingly kills Frieza. Goku and Frieza’s fight drags on suuuuuper long.
From there, everyone’s back on earth and Vegeta’s alive and being a dick again. he thinks since Goku and Frieza are dead he’s the most powerful warrior alive now (he has a decent case tbf), except BAM frieza shows back up on earth, in his new robot form with his dad to back him up (like i told you he’s prone to hissy fits). so everyone’s REALLY pooping their pants BIG TIME now, but out of NO WHERE some new super saiyan (remember they’re all supposed to be dead?!). This guy’s powerful as a fresh set of ovaries and proceeds to absolutely dee-STROY frieza with a sweet-ass sword. This guy turns out to be from the future, his name is Trunks and he’s Vegeta’s son. Trunks is arguably the coolest guy in the show from this point on. Trunks tells them some androids are gonna show up soon and destroy the planet because Goku dies of heart disease and can’t stop them. He gives Goku some medicine and time travels away. The gang train solidly for three years until the androids are supposed to show. They do, and this is where things start to get really complicated, so i’m gonna try blitz through it:
goku dies in the battle though, then spends the rest of the series fighting tournaments in the afterlife. some stuff happens with a guy who turns goodies evil, and then there’s GT which i dont even know anything about. hope that helps!