While Roman emperors were empowered to choose their own successor, the first emperor to actually be succeeded by his own natural-born son was Vespasian.
If you want a good overview of the early Roman emperors, beginning with Augustus (adopted son of Julius Caesar), then I strongly recommend the BBC mini-series I, Claudius. It is not 100% historically accurate, but it’s great fun and includes the miraculous sight of Patrick Stewart with hair. Brian Blessed, Derek Jacobi, Siân Phillips, John Hurt, and John Rhys-Davies also knock it out of the park. What the series drives home is that there are few things more dangerous than being a Roman emperor.
Augustus died of natural causes (apparently), and was succeeded by his third wife’s son by another marriage. That son, Tiberius, was possibly murdered; he was succeeded by his great-nephew Caligula. Caligula was… not popular. He was murdered by some senators and his own guards, and succeeded by his uncle Claudius. Claudius was also murdered, by his fourth wife. Her son was the next emperor: Nero.
Nero was not a fantastic guy either – noticing a trend yet? Nero committed suicide (“what an artist dies with me!”) and left a power vacuum. Four emperors took power in rapid succession: Galba, Otho, Vitellius, and Vespasian. All but Vespasian fell to murder or suicide. Vespasian, miraculously, survived a good nine years before dying of natural causes… and he was the first emperor to actually get his own son into power.
That son, Titus, lasted for two years before also dying of natural causes. He was succeeded by his brother Domitian. Who was assassinated, breaking the streak. Oh well.
End note: some of Vespasian’s last words were “Dear me, I think I’m becoming a god,” proving that if there’s one thing all the Roman emperors had in common, it’s hubris.