Vespasian finally left West Asia and sailed to Rome in the spring of 70 AD, when it was safe to travel on the Mediterranean, after the winter storms were over. He turned out to be a pretty good emperor, though he was always more interested in the army than anything else, and he favored the army of West Asia that he had commanded. He was popular because he lived very simply, and didn't throw big parties or spend a lot of money like the Julio-Claudian emperors. So the empire had a lot of money in the treasury. Vespasian was emperor for ten years, until he died in bed in 79 AD. When he died, he is supposed to have said "Oh my God, I think I'm becoming a god!"
When Vespasian died, his older son Titus took over. Everyone was happy to avoid another civil war. Almost immediately Titus had to deal with the volcanic eruption that buried Pompeii. Soon Titus ran into some trouble with the Senate because he fell in love with a Jewish woman named Berenice, who was a princess in West Asia, but not Roman. Titus also crushed a Jewish revolt in Israel and destroyed the big Jewish temple in Jerusalem, which he reminded Romans about with a big stone triumphal arch. Still he is mostly remembered as a good emperor. He died young, in 81 AD, after ruling less than three years.
Titus had no sons, so Titus' younger brother Domitian became emperor. Domitian was a very different sort of man. He had always felt that his father, Vespasian, liked Titus better, and this feeling made him angry and mean. He even tried to organize revolts against Titus when Titus was emperor. As emperor, Domitian was convinced that everyone was plotting to kill him, and he had many senators and other people killed because he suspected them. He also may have persecuted some early Christians. He also made people call him "Lord and God" (Dominus et Deus). In the end people couldn't stand this sort of behavior, and he was really assassinated in 96 AD. (This story has something in common with the story of the Athenian tyrant Hippias, doesn't it?)
For more on the Flavians, check out these books from Amazon.com or from your library:
Classical Rome, by John Clare (1993). For kids, the whole political history from beginning to end.
The Romans: From Village to Empire, by Mary Boatwright, Daniel Gargola, and Richard Talbert (2004). Okay, it's a little dry, but it is up to date and has all the facts you could want.
The Roman Empire, by Colin Wells (1984). More readable. Alternates chapters on political and social history. Unfortunately, he stops at the third century crisis.
Vespasian, by Barbara Levick (1999). The only biography of Vespasian in English. Mainly for academics.
The Emperor Domitian, by Brian Jones (1992). Makes Domitian seem like not such a bad guy. Jones also has written a biography of Titus, but it's out of print.
Suetonius: The Flavian Emperors : A Historical Commentary, by Brian Jones and Robert Milns (2003). Academic commentary on the Roman historian Suetonius' account of these emperors.