Everybody was tired of all this fighting and killing. And everyone was afraid of Octavian. So when Octavian just kept ruling almost as if he were king, nobody tried to stop him. He made people call him Augustus (which means The Good) instead of Octavian. But he was smarter than his uncle Julius Caesar had been.
Augustus didn't call himself dictator, but First Citizen. He didn't disband the Senate; he made the Senate do what he wanted. He had himself elected tribune, so he could veto whatever the Senate did that he didn't like. People knew that Augustus was really taking over, but as long as there was peace and he didn't SAY he was taking over, it was okay with them.
Augustus' wife Livia
Augustus lived a long time, until 14 AD. When he died, hardly anyone could remember before he was in power, or all they remembered was killing and blood. His son-in-law (his daughter Julia's husband) Tiberius took over as First Citizen. Tiberius wasn't really a very good ruler (we call them emperors now, but they didn't call themselves that). He spent a lot of time swimming and having big parties. He left most of the work to his assistants. But still people thought that was better than civil war.
Tiberius died in 37 AD, and his nephew Germanicus' son Gaius, who is often called Caligula, took over. Caligula wasn't too bad at first, but he seems to have suffered from mental illness. After a while he started doing things like trying to make his horse a senator, and trying to marry his sister. By 41 AD people decided he was too hard to deal with and his own guards killed him.
For more on the Julio-Claudians, check out these books from Amazon or from your library:
Classical Rome, by John Clare (1993). For kids, the whole political history from beginning to end.
Oxford First Ancient History, by Roy Burrell (reissued 1997). For kids. It skips around a lot, not trying to tell everything, just highlights.
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 Revolution, by Ronald Syme (1960). Still a classic.
From the Gracchi to Nero: A History of Rome from 133 B.C. to A.D. 68
by H. H. Scullard (1959, 5th edition 1990). Another classic.