Julian was the last surviving male relative of Constantine, so it wasn't clear who should become the next emperor after he was killed. The generals in the East got together in 363 AD and chose one of themselves, a Christian named Jovian. Jovian gave in to the Sassanids, and retreated, but he didn't last very long: he died in 364 AD.
Jovian was succeeded by another Eastern general, Valentinian, who was a Catholic. Valentinian soon decided that he would take control of the western part of the Roman Empire and leave the eastern part to his brother Valens. Valentinian's capital was at Milan, while Valens, an Arian, lived in Constantinople.
Valens' young son Gratian was engaged to Constantius II's daughter Constantia, to make the new emperors part of the old family, and in 367 AD he was made emperor as well, though he was only eight years old! Gratian's teacher was the Christian Ausonius, but we know more about these people from Ammianus Marcellinus.
In this period the Romans continued to fight both the Germans and the Sassanids. There were also Donatist rebellions in Africa. In 378 AD the Romans lost an important battle at Adrianople (the city of Hadrian) in the Balkans, where Valens was killed fighting the Visigoths. The Visigoths pushed their way into the Roman Empire and settled down, and the Romans gave them refugee status and let them stay.
Theodosius, a young Catholic general of Spanish origin, and the son of another general, was chosen to succeed Valens. He ruled along with Gratian and Valentinian II, the young son of Valentinian I, and he married Galla, the daughter of Valentinian I. Theodosius was able to regain the upper hand militarily, though mostly by making treaties with both the Visigoths and the Sassanids. In 383, a rebellious general named Maximus killed Gratian, and in 388 the same man attacked Valentinian II; Theodosius then killed Maximus.
In 391 AD, under the influence of Bishop Ambrose (now St. Ambrose) of Milan, Theodosius made it illegal to practice pagan religious rituals even in your own house. Only Christianity (and to some extent Judaism) were now legal in the Roman Empire. A non-Roman general named Arbogast rebelled at this news, killed Valentinian II and put a Roman friend of his, Eugenius, on the throne, but Theodosius defeated and killed them. By the time he died in 395 AD, he left a firmly Christian empire.
The Ancient Roman World, by Ronald Mellor (2004). Straight political history, For teens.
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.
Theodosius: The Empire at Bay, by Stephen Williams and Gerard Friell (1998).