Goku Son
In this series, Goku is now an adult. He is married to Chichi to whom he'd made a promise of marriage in his youth before he'd really known what it even meant. Chichi, however, had taken the vow seriously and went hunting for him. In Dragonball Z, they have had a child which they named after Goku's adopted grandfather -- Gohan.
Gohan
Gohan begins the series as a 5 year old and it follows his adventures as well as that of his father's until he reaches adulthood. While trying to live up to his potential as Goku's son and become a force to protect the Earth, Gohan is also pulled by his mother to excel in academics. He tries his best to do both.
Bulma
Bulma and the others still keep in close contact with one another. As in times past, her Dragon Radar proves extremely useful when they need to locate the Dragonballs as well as the help she and her father gives them when the group has need of a working space ship.
Now that she is grown up, Bulma will eventually involve herself with Vegita, a prince belonging to the Saiyan race. Her son Trunks will take after his father in skill, but have his mother's good looks and smarts.
Picolo
Picolo has set aside his usual meaness to help Gohan train in the skills he will need to have to protect the Earth after his father is killed(this is only temporary, but it will still take Goku a year to make his way back). Though always wanting to appear uncaring, Picolo cames to care very much for Gohan even at the cost of his own life.
Krillin
A close friend to the Son family, Krillin will become one of the strongest earthlings on the planet. He like Picolo, becomes very fond of Gohan and tries to protect him in any way he can.
Dragonball Z picks up 5 years after Goku's mariage to Chichi. The focus changes from Goku to his son Gohan. The series also becomes more involved with extra terrestials as Goku's race the Saiyans, find out where he has gone. After Goku gives up his life defending his son from the Saiyan who has come to claim him and the power of the Dragonballs, Gohan is trained by Picolo to be able to defend himself and possibly the Earth from the Saiyans they expect to come after the first. Meanwhile Goku trains in heaven in order to be able to return to Earth and does so in time to meet Vejita, Prince of the Saiyan people.
Vejita is defeated and his heart forever changed by his encounter on Earth with Goku and Gohan. It is then that we find out that Goku's world has been destroyed by the evil Freezer and that he and Vejita (who is a prince) are the last of their kind. It is now that the adventures move out to the world where Kami-sama and Picolo's people come from --Namik.
After the threat of Freeza is done with, Goku is seemingly lost as the Namik planet explodes. Unable to bring him back with the use of the Namiks more powerful Dragonballs, Gohan and the others must try and protect the planet themselves from the threat of Goku's old enemies Red Ribbon. After Goku is able to return to them, they must then deal with the terror of Cell, an artificially constructed death machine, and later the demon Boo.
As in Dragonball, you can't help but have these people become a part of your lives, their troubles becoming your troubles, their fights your fights. Though not as intersperced with humor as Dragonball, Dragonball Z still tends to throw some in when least expected.